I invite you to read this letter as a good example in context. Note how he treats the judgement of the Catholic Church and tradition handed down concerning scriptural faith.(source of SP's statement)
For what is so manifestly shown to be evil, it is not necessary to waste time in exposing further, lest contentious persons think the matter doubtful. It is enough merely to answer such things as follows: we are content with the fact that this is not the teaching of the Catholic Church, nor did the fathers hold this.
Athanasius' statement that "we are content with the fact that this is not the teaching of the Catholic Church, nor did the fathers hold this" doesn't explicitly rely on Scripture, which is doubtless why it is being cited.
Standing alone, however, they don't directly contradict the idea that Scripture is formally sufficient. Instead, this kind of comment says that it is enough that the church as a whole doesn't teach this and that there is no historical precedent for the teaching.
One might argue that by "the teaching of the Catholic Church" Athanasius is referring to Nicaea. After all, the letter begins, immediately after the greeting sentence:
I thought that all vain talk of all heretics, many as they may be, had been stopped by the Synod which was held at Nicæa.- Athanasius, Letter 59, Section 1 (source)
In fact, that sentence in isolation also looks nothing like a view of formal sufficiency of Scripture (although it doesn't actually interact with the view of formal sufficiency).
Likewise, a couple of sentences in the conclusion don't explicitly mention Scripture:
But thanks to the Lord, much as we were grieved at reading your memoranda, we were equally glad at their conclusion. For they departed with concord, and peacefully agreed in the confession of the pious and orthodox faith.- Athanasius, Letter 59, Section 12 (source)
However, when we begin to examine Athanasius' comments in context, a different picture emerges. Here's the entire Section 3 from which the initial comment was drawn:
Such were the contents of the memoranda; diverse statements, but one in their sense and in their meaning; tending to impiety. It was for these things that men who make their boast in the confession of the fathers drawn up at Nicæa were disputing and quarrelling with one another. But I marvel that your piety suffered it, and that you did not stop those who said such things, and propound to them the right faith, so that upon hearing it they might hold their peace, or if they opposed it might be counted as heretics. For the statements are not fit for Christians to make or to hear, on the contrary they are in every way alien from the Apostolic teaching. For this reason, as I said above, I have caused what they say to be baldly inserted in my letter, so that one who merely hears may perceive the shame and impiety therein contained. And although it would be right to denounce and expose in full the folly of those who have had such ideas, yet it would be a good thing to close my letter here and write no more. For what is so manifestly shewn to be evil, it is not necessary to waste time in exposing further, lest contentious persons think the matter doubtful. It is enough merely to answer such things as follows: we are content with the fact that this is not the teaching of the Catholic Church, nor did the fathers hold this. But lest the ‘inventors of evil things [Rom. i. 30.]’ make entire silence on our part a pretext for shamelessness, it will be well to mention a few points from Holy Scripture, in case they may even thus be put to shame, and cease from these foul devices.- Athanasius, Letter 59, Section 3 (source)
Notice that Athanasius says: "For the statements are not fit for Christians to make or to hear, on the contrary they are in every way alien from the Apostolic teaching." That Apostolic teaching, of course, is Scripture. For Athanasius, it is adherence to the Apostolic teaching that is the guiding principle.
Notice as well that Athanasius doesn't view the error of the heretics as being at all reasonable. He thinks that the bare repetition of their argument is enough to show its impiety, "For what is so manifestly shewn to be evil, it is not necessary to waste time in exposing further, lest contentious persons think the matter doubtful." That middle expression, "it is not necessary to waste time in exposing further," is then explained by the comment about the teaching not being one of the Catholic Church or the fathers. It is also a reiteration of his prior comment, in Section 2, "I write this after reading the memoranda submitted by your piety, which I could wish had not been written at all, so that not even any record of these things should go down to posterity. For who ever yet heard the like? Who ever taught or learned it?"
However, notice that the way in which these men are to be put to shame and converted from their evil way is not a simple appeal to Nicaea (after all, these men "make their boast in the confession of the fathers drawn up at Nicæa") nor to the fathers (from what fathers does Athanasius quote?) but from Scripture: "But lest the ‘inventors of evil things [Rom. i. 30.]’ make entire silence on our part a pretext for shamelessness, it will be well to mention a few points from Holy Scripture, in case they may even thus be put to shame, and cease from these foul devices."
The same contextual issues arise with respect to the introductory section of Athanasius' letter. Looking at the context, indeed the next sentence, we see a somewhat different sentiment:
I thought that all vain talk of all heretics, many as they may be, had been stopped by the Synod which was held at Nicæa. For the Faith there confessed by the Fathers according to the divine Scriptures is enough by itself at once to overthrow all impiety, and to establish the religious belief in Christ.- Athanasius, Letter 59, Section 1 (source)
Notice that Athanasius is quick to point out the rule that guided the fathers at Nicaea, namely the Scriptures. While there is some mention in the text of holding to the faith of the Nicene fathers, the bulk of the letter is exegetical - relying on and explaining the matter from Scripture.
So we should not be surprised at the conclusion mentioned above considered in its context:
This proves that while to all the others the Word came, in order that they might prophesy, from Mary the Word Himself took flesh, and proceeded forth as man; being by nature and essence the Word of God, but after the flesh man of the seed of David, and made of the flesh of Mary, as Paul said [Cf. Rom. i. 3; Gal. iv. 4.]. Him the Father pointed out both in Jordan and on the Mount, saying, ‘This is My beloved Son in whom I am well pleased [Matt. iii. 17, and xvii. 5.].’ Him the Arians denied, but we recognizing worship, not dividing the Son and the Word, but knowing that the Son is the Word Himself, by Whom all things are made, and by Whom we were redeemed. And for this reason we wonder how any contention at all has arisen among you about things so clear. But thanks to the Lord, much as we were grieved at reading your memoranda, we were equally glad at their conclusion. For they departed with concord, and peacefully agreed in the confession of the pious and orthodox faith. This fact has induced me, after much previous consideration, to write these few words; for I am anxious lest by my silence this matter should cause pain rather than joy to those whose concord occasions joy to ourselves. I therefore ask your piety in the first place, and secondly those who hear, to take my letter in good part, and if anything is lacking in it in respect of piety, to set that right, and inform me. But if it is written, as from one unpractised in speech, below the subject and imperfectly, let all allow for my feebleness in speaking.- Athanasius, Letter 59, Section 12 (source)
Notice that Athanasius writes: "And for this reason we wonder how any contention at all has arisen among you about things so clear." These are not vague obscure things, but clear things. And the thing from which this is clear can be seen from the proof that Athanasius had provided in, for example, Section 5 of his letter:
. But this is not so, far be the thought. For he ‘takes hold of the seed of Abraham [Heb. ii. 16.],’ as the apostle said; whence it behoved Him to be made like His brethren in all things, and to take a Body like us. This is why Mary is truly presupposed, in order that He may take it from her, and offer it for us as His own. And this Isaiah pointed to in his prophecy, in the words: ‘Behold the Virgin [Isa. vii. 14.],’ while Gabriel is sent to her—not simply to a virgin, but ‘to a virgin betrothed to a man [Luke i. 27.],’ in order that by means of the betrothed man he might shew that Mary was really a human being. And for this reason Scripture also mentions her bringing forth, and tells of her wrapping Him in swaddling clothes; and therefore, too, the paps which He sucked were called blessed [Ib. xi. 27.]. And He was offered as a sacrifice, in that He Who was born had opened the womb [Ib. ii. 23.]. Now all these things are proofs that the Virgin brought forth. And Gabriel preached the Gospel to her without uncertainty, saying not merely ‘what is born in thee,’ lest the body should be thought to be extraneously induced upon her, but ‘of thee,’ that what was born might be believed to be naturally from her, inasmuch as Nature clearly shews that it is impossible for a virgin to produce milk unless she has brought forth, and impossible for a body to be nourished with milk and wrapped in swaddling clothes unless it has previously been naturally brought forth. This is the meaning of His being circumcised on the eighth day: of Symeon taking Him in his arms, of His becoming a young child, and growing when He was twelve years old, and of His coming to His thirtieth year. For it was not, as some suppose, the very Essence of the Word that was changed, and was circumcised, because it is incapable of alteration or change. For the Saviour Himself says, ‘Behold, behold, it is I, and I change not [Mal. iii. 6.],’ while Paul writes: ‘Jesus Christ, the same yesterday, and to-day, and for ever [Heb. xiii. 8.].’ But in the Body which was circumcised, and carried, and ate and drank, and was weary, and was nailed on the tree and suffered, there was the impassible and incorporeal Word of God. This Body it was that was laid in a grave, when the Word had left it, yet was not parted from it, to preach, as Peter says, also to the spirits in prison [1 Pet. iii. 19.].- Athanasius, Letter 59, Section 5 (source)
Notice how Athanasius explains it as being that "Nature clearly shews that it is impossible for a virgin to produce milk unless she has brought forth, and impossible for a body to be nourished with milk and wrapped in swaddling clothes unless it has previously been naturally brought forth." It's not a matter of obscurity or ambiguity or one man's opinion over another. It is clear and plain to Athanasius.
Notice as well that after more Scriptural exegesis in sections 6-7, Athanasius provides the following comments in section 8:
These things being thus demonstrated, it is superfluous to touch upon the other points, or to enter upon any discussion relating to them, since the body in which the Word was is not coessential with the Godhead, but was truly born of Mary, while the Word Himself was not changed into bones and flesh, but came in the flesh. For what John said, ‘The Word was made flesh [Joh. i. 14.],’ has this meaning, as we may see by a similar passage; for it is written in Paul: ‘Christ has become a curse for us [Gal. iii. 13.].’ And just as He has not Himself become a curse, but is said to have done so because He took upon Him the curse on our behalf, so also He has become flesh not by being changed into flesh, but because He assumed on our behalf living flesh, and has become Man. For to say ‘the Word became flesh,’ is equivalent to saying ‘the Word has become man;’ according to what is said in Joel: ‘I will pour forth of My Spirit upon all flesh [Joel ii. 28.];’ for the promise did not extend to the irrational animals, but is for men, on whose account the Lord is become Man. As then this is the sense of the above text, they all will reasonably condemn themselves who have thought that the flesh derived from Mary existed before her, and that the Word, prior to her, had a human soul, and existed in it always even before His coming. And they too will cease who have said that the Flesh was not accessible to death, but belonged to the immortal Nature. For if it did not die, how could Paul deliver to the Corinthians ‘that Christ died for our sins, according to the Scriptures [1 Cor. xv. 3.],’ or how did He rise at all if He did not also die? Again, they will blush deeply who have even entertained the possibility of a Tetrad instead of a Triad resulting, if it were said that the Body was derived from Mary. For if (they argue) we say the Body is of one Essence with the Word, the Triad remains a Triad; for then the Word imports no foreign element into it; but if we admit that the Body derived from Mary is human, it follows, since the Body is foreign in Essence, and the Word is in it, that the addition of the Body causes a Tetrad instead of a Triad.- Athanasius, Letter 59, Section 8 (source)
Of note, as a first point, is the fact that Athanasius begins the section by again repeating that even all this additional evidence he is giving is not necessary. Nevertheless, he's providing it.
Also, of significance, notice that Athanasius teaches us (by example) to use Scripture to interpret Scripture when he states: "For what John said, ‘The Word was made flesh [Joh. i. 14.],’ has this meaning, as we may see by a similar passage;" which shows us that Athanasius agreed with us that Scripture is to be Scripture's interpreter. (He does this again almost immediately: "For to say ‘the Word became flesh,’ is equivalent to saying ‘the Word has become man;’ according to what is said in Joel: ‘I will pour forth of My Spirit upon all flesh [Joel ii. 28.];’")
Notice as well that Athanasius thinks that the sense of Scripture is so clear that the heretics will themselves see their error and condemn themselves. For he writes: "As then this is the sense of the above text, they all will reasonably condemn themselves who have thought that the flesh derived from Mary existed before her, and that the Word, prior to her, had a human soul, and existed in it always even before His coming."
In section 9, Athanasius provides an interesting comment. He writes:
And how do these remain Christians who imagine another God in addition to the true one? For, once again, in their other fallacy one can see how great is their folly. For if they think because it is contained and stated in the Scriptures, that the Body of the Saviour is human and derived from Mary, that a Tetrad is substituted for a Triad, as though the Body created an addition, they go very far wrong, so much so as to make the creature equal to the Creator, and suppose that the Godhead can receive an addition.- Athanasius, Letter 59, Section 9 (source)
Notice that Athanasius assumes that these putative Christians will derive their understanding from Scripture. Accordingly, he takes care to state their their understanding is wrong (as well as, at some length, to explain why it is wrong).
Athanasius' argument reaches something of a pinnacle when, in section 10, he provides a very plain Scriptural demonstration that he thinks they must acquiesce to:
For this reason they also will henceforth keep silence, who once said that He who proceeded from Mary is not very Christ, or Lord, or God. For if He were not God in the Body, how came He, upon proceeding from Mary, straightway to be called ‘Emmanuel, which is being interpreted God with us [Matt. i. 23.]?’ Why again, if the Word was not in the flesh, did Paul write to the Romans ‘of whom is Christ after the flesh, Who is above all God blessed for ever. Amen [Rom. ix. 5.]?’ Let them therefore confess, even they who previously denied that the Crucified was God, that they have erred; for the divine Scriptures bid them, and especially Thomas, who, after seeing upon Him the print of the nails, cried out ‘My Lord and my God [John xx. 28.]!’ For the Son, being God, and Lord of glory [1 Cor. ii. 8.], was in the Body which was ingloriously nailed and dishonoured; but the Body, while it suffered, being pierced on the tree, and water and blood flowed from its side, yet because it was a temple of the Word was filled full of the Godhead. For this reason it was that the sun, seeing its creator suffering in His outraged body, withdrew its rays and darkened the earth. But the body itself being of mortal nature, beyond its own nature rose again by reason of the Word which was in it; and it has ceased from natural corruption, and, having put on the Word which is above man, has become incorruptible.- Athanasius, Letter 59, Section 10 (source)
It's particularly remarkable to see how Athanasius portrays Scripture as commanding them to believe ("for the divine Scriptures bid them"). Athanasius recognizes that the Scriptures can bid one to believe - they have authority - and they are the indisputable rule of faith. Athanasius indicates that the Christians will silence their own error once they see what Scripture so plainly teaches.
Section 11, which is about the only section we haven't discussed, continues the same Scriptural arguments:
But with regard to the imagination of some, who say that the Word came upon one particular man, the Son of Mary, just as it came upon each of the Prophets, it is superfluous to discuss it, since their madness carries its own condemnation manifestly with it. For if He came thus, why was that man born of a virgin, and not like others of a man and woman? For in this way each of the saints also was begotten. Or why, if the Word came thus, is not the death of each one said to have taken place on our behalf, but only this man’s death? Or why, if the Word sojourned among us in the case of each one of the prophets, is it said only in the case of Him born of Mary that He sojourned here ‘once at the consummation of the ages [Heb. ix. 26.]?’- Athanasius, Letter 59, Section 11 (source)
Notice how Athanasius refers to the opinion of men who suggest that the Word came to Jesus, rather than Jesus being the Word made flesh as "madness" in view of the plain Scriptural testimony. (As an aside, it is interesting to note that Athanasius viewed only Christ's death as having taken place on our behalf, and not also the deaths of the martyrs.)
In conclusion, there are indeed passages of Athanasius that don't expressly rely on Scripture. For example, in section 4:
For they say that God came in a human body. But the fathers who also assembled at Nicæa say that, not the body, but the Son Himself is coessential with the Father, and that while He is of the Essence of the Father, the body, as they admitted according to the Scriptures, is of Mary. Either then deny the Synod of Nicæa, and as heretics bring in your doctrine from the side; or, if you wish to be children of the fathers, do not hold the contrary of what they wrote.- Athanasius, Letter 59, Section 4 (source)
But when we see the fact that the fathers of Nicaea are mentioned only occasionally, while Scripture is mentioned throughout and that even comments like the comment above from section 4 is immediately preceded by the following sentences:
Whence did it occur to you, sirs, to say that the Body is of one Essence with the Godhead of the Word? For it is well to begin at this point, in order that by shewing this opinion to be unsound, all the others too may be proved to be the same. Now from the divine Scriptures we discover nothing of the kind.- Athanasius, Letter 59, Section 4 (source)
In short, for Athanasius it is those divine Scriptures that are the rule of faith, and consequently they are what the fathers at Nicaea followed and taught. As Athanasius put it in section 1, "the Faith there confessed by the Fathers according to the divine Scriptures ... ."