This argument against private judgment can take various forms. One form of the argument is a syllogism in the form:
1. If God gave us a way to know the truth, that way would give us knowledge of the truth with reasonable certainty.
2. Private judgment doesn't provide reasonable certainty, because reasonable people differ in the application of private judgment.
3. Therefore, private judgment is not the way God gave us to know the truth.
There are several problems with this argument. The number one problem is that it employs the fallacy of skepticism. The way that it employs the fallacy of skepticism is in establishing the minor premise, i.e. private judgment doesn't provide reasonable certainty.
This is a logical fallacy for a couple of reasons. The most obvious reason is that private judgment is necessarily used to deny that private judgment provides reliable conclusions. If the conclusion is correct (i.e. that private judgment does not provide reliable conclusions) then the conclusion itself is not reliable since it obtained by private judgment.
Second, this is a logical fallacy in the sense of simply being a universal denial of knowability of information. That is to say, this argument lacks uniqueness. It is not particularly a criticism of the "Protestant" position. It is a modus tolens argument that can be applied mutatis mutandis to any epistemology.
Specifically, this same argument can be applied to the epistemology of Catholicism, because Catholicism too requires, at some link in the chain, one to use private judgment. This is illustrated in the attempt of certain of Rome's apologists to escape the apparent circularity of Rome's epistemology.
The apparent circularity is this:
1. The Bible is right because the Church says it is.
2. The Church is right because the Bible says it is.
There, the circularity is obvious. It is bigger than the circle of:
1. The Bible is right because the Bible says it is.
... but it is still a circle.
To try to escape this circle, some apologists for Catholicism use what they call a "spiral argument" that was apparently developed by Karl Keating, one of the more prominent apologists for Catholicism (although I cannot recall him debating anyone from the Reformed side of the Tiber river in a long time).
Here's one presentation of the argument:
A Spiral Argument(link to source) (Notice how, unlike Mr. Patrick Madrid, we're not afraid to let the reader see the writings to which we're responding.)
Note that this is not a circular argument. We are not basing the inspiration of the Bible on the Church’s infallibility and the Church’s infallibility on the word of an inspired Bible. That indeed would be a circular argument! What we have is really a spiral argument. On the first level we argue to the reliability of the Bible insofar as it is history. From that we conclude that an infallible Church was founded. And then we take the word of that infallible Church that the Bible is inspired. This is not a circular argument because the final conclusion (the Bible is inspired) is not simply a restatement of its initial finding (the Bible is historically reliable), and its initial finding (the Bible is historically reliable) is in no way based on the final conclusion (the Bible is inspired). What we have demonstrated is that without the existence of the Church, we could never know whether the Bible is inspired.
Let's assess this argument. The core of the argument is:
1. [W]e argue to the reliability of the Bible insofar as it is history.
2. From that we conclude that an infallible Church was founded.
3. And then we take the word of that infallible Church that the Bible is inspired.
We could reasonably expand this argument to the following:
1. The Bible is an historically reliable document.
2. The Bible records the founding of a church.
3. The Bible indicates that this church is infallible.
4. This infallible church is the church headed by the pope.
5. This infallible church teaches that the Bible is inspired.
It could probably be expanded even further, but this is enough for the purposes of illustrating the problems with this supposedly spiral argument.
Problem 1: Private Judgment Vastly Multiplied
Rather than simply accepting the Bible as Inspired Word of God based on one private judgment, this "spiral" argument requires one to employ private judgment over and over and over again.
First, one uses private judgment to answer the question of historical reliability.
Second, one uses private judgment to decide the meaning of Scripture as to whether a single, institutionally unitary church or many institutionally separate churches were founded.
Third, one uses private judgment to decide that this single church is taught as being infallible rather than as being fallible.
Fourth, one uses private judgment to identify this single church as the Roman Catholic church instead of, say, the Eastern Orthodox church or the Ethiopian Orthodox Church.
Fifth, one uses private judgment to decide that the Roman Catholic church teaches that the Bible is inspired, as opposed to teaching that the Bible simply contains God's word.
This is a vast multiplication of private judgment over, for example, simply accepting the Bible on historical grounds and then accepting that the Bible says that the Bible is inspired, or simply accepting the Bible's claim of inspiration as true in the first and only step of the process.
If private judgment is inherently bad, the "spiral argument" uses more of it.
Problem 2: Bootstrapping Error
The Spiral argument attempts to avoid the obvious "jump" from nothing to accepting the Bible's claim of inspiration by steps. But each (or at least several) of the steps are "jumps" in themselves.
There is a jump from nothing to accepting the historical method as providing reliable conclusions.
There is a jump from using the historical method to confirm the general reliability of the Bible, to accepting a particular historical account in the Bible.
There is a jump from accepting an account as historical to accepting the doctrine taught in the event as truth.
There is a jump from accepting the general idea that a church was founded to accepting that a particular church is that church.
If making jumps is bad, breaking up a big jump into several smaller jumps doesn't solve the problem, it just distributes it.
Problem 3: Inspiration Smuggled Back In
In fact, the "spiral argument" is circular, because inspiration is smuggled into step 3 of my expanded formulation of the argument, or step 2 of the original formulation of the argument. That is to say, the teaching of an "infallible church" is accepted allegedly because the Bible as a historically accurate document is accepted. But the claim of infallibility is a theological claim, not an historical claim, and it is accepted either because of the authority of the Bible or the person speaking in the Bible.
Problem 4: Church Infallibility Smuggled Back In
Furthermore, the "spiral argument" has a second circularity, in that the infallibility of the church is smuggled back into the argument twice. It is smuggled back once in telling people which church to accept as "the church," and again (more importantly) in interpreting Scripture as teaching an infallible church in the first place.
Specifically, the claim that the Scriptures disclose the founding of "an infallible church" requires loads of eisegesis - of reading into the text, rather than of obtaining teachings from the text.
Problem 5: Scripture Promotes Private Judgment
Worse (for Catholicism) than the issue simply being a matter of silence, Scripture actually encourages the use of private judgment. For example, the Scripture many times and in various ways encourages people to apply personal judgment to arrive at the truth. For example, the Bereans are commended for using private judgment and Scripture to judge Paul, and Paul tells Timothy that the Scriptures are able to make one "wise unto salvation." Furthermore, John tells us that his gospel was written so that we would believe it and have life through faith in Jesus the Messiah, the Son of God.
So, next time a person tries to tell you that the difference between Protestant interpretation and Romanist interpretation is that the Protestant makes himself his own ultimate authority, be prepared to challenge that deceptive claim. Everyone uses private judgment. If private judgment is inherently untrustworthy, the Roman position is actually worse off than the "Protestant" position.
Furthermore, while it might be nice to hand over one's brain to the church, so that one doesn't have to think about the meaning of Scripture, that's just not how God ordained things. The fact that it would be convenient or handy doesn't make it so.
Instead, God provided fallible churches with fallible elders over them. These fallible teachers are to teach the Scriptures to their people, but the unchanging Scriptures serve as the rule and measure of the Christian faith, with the fallible churches serving as guides.