Do you mean that the depraved heart CAN believe, or just doesn't want
The impasse we are at may be due to what you mean by "CAN" and what I mean be "CAN."Let me see if I can find a mountain pass.
Men make choices. Choices are not random - they have a reason and a motive. We can see this in Scripture, even in God's choices. This reason and motive is the cause of the choice. We know that motive is critical to people's choices, and that's why - if you've ever watched a crime show, or read a detective novel - you'll see the detectives looking for the killer's motive. We all know that people make choices based on motives and reasons. We accept that as a fact of life. If someone wants to defend himself from a crime, he can tell a jury "But what would my motive be for doing that?" If there is no motive, most people will be reluctant to say that the man in the dock was really the killer.
So, anyway, Scripture and common sense agree that choices have reasons, causes, and motives.
Furthermore, we can see that the way that man's will works is that man chooses what seems best to him. Consider our mother Eve. She saw that the fruit of the tree was good to eat, and was desireable because it brought wisdom. That was the reason and motive for her eating the forbidden fruit. She was wrong in her assessment of what was best. She did not give appropriate weight to God's command, and eating seemed best to her. So she chose to eat.
Why is that the motive and reason of wisdom-seeking and nutrition outweighed the motive and reason of obedience? Eve was foolish: the answer is Eve's heart.
It is important to note, here, that Eve was not depraved, she was naive and gullible. Nevertheless, while man now has a depraved nature that provides a corrupt motive, his will works in much the same way - choosing according to motive.
For Scripture says:
34O generation of vipers, how can ye, being evil, speak good things? for out of the abundance of the heart the mouth speaketh. 35A good man out of the good treasure of the heart bringeth forth good things: and an evil man out of the evil treasure bringeth forth evil things.
18But those things which proceed out of the mouth come forth from the heart; and they defile the man. 19For out of the heart proceed evil thoughts, murders, adulteries, fornications, thefts, false witness, blasphemies:
Mark 7:21 For from within, out of the heart of men, proceed evil thoughts, adulteries, fornications, murders,
Luke 6:45 A good man out of the good treasure of his heart bringeth forth that which is good; and an evil man out of the evil treasure of his heart bringeth forth that which is evil: for of the abundance of the heart his mouth speaketh.
Some of those verses show the contrast between a good heart and and evil heart, but one more is helpful:
2 Timothy 2:22 Flee also youthful lusts: but follow righteousness, faith, charity, peace, with them that call on the Lord out of a pure heart.
Yet, as you recall:
Jeremiah 17:9 The heart is deceitful above all things, and desperately wicked: who can know it?
How does the heart change from desparately wicked to pure?
Scripture says that God changes, purifies, and cleanses the heart:
2 Chronicles 30:12 Also in Judah the hand of God was to give them one heart to do the commandment of the king and of the princes, by the word of the LORD.
Jeremiah 24:7 And I will give them an heart to know me, that I am the LORD: and they shall be my people, and I will be their God: for they shall return unto me with their whole heart.
Jeremiah 32:39 And I will give them one heart, and one way, that they may fear me for ever, for the good of them, and of their children after them:
Ezekiel 11:19 And I will give them one heart, and I will put a new spirit within you; and I will take the stony heart out of their flesh, and will give them an heart of flesh:
Ezekiel 36:26 A new heart also will I give you, and a new spirit will I put within you: and I will take away the stony heart out of your flesh, and I will give you an heart of flesh.
Acts 16:14 And a certain woman named Lydia, a seller of purple, of the city of Thyatira, which worshipped God, heard us: whose heart the Lord opened, that she attended unto the things which were spoken of Paul.
James 4:8 Draw nigh to God, and he will draw nigh to you. Cleanse your hands, ye sinners; and purify your hearts, ye double minded.
And again, lest we think that James is saying that man can do this themselves:
Proverbs 20:9 Who can say, I have made my heart clean, I am pure from my sin?
Matthew 5:8 Blessed are the pure in heart: for they shall see God.
1 Timothy 1:1-5
1Paul, an apostle of Jesus Christ by the commandment of God our Saviour, and Lord Jesus Christ, which is our hope; ... 5 Now the end of the commandment is charity out of a pure heart, and of a good conscience, and of faith unfeigned:
Thus, we see that man's choices come from his heart: bad choices from a bad heart, and good choices from a good heart. How does a bad heart become good? By God doing something to it.
A lot of people cling to James 4, thinking that James' command means that we can purify our own hearts. Of course, comparing Scripture with Scripture, we see that such is not the case. For we don't just have Proverbs (quoted above), but indeed James himself states:
17Every good gift and every perfect gift is from above, and cometh down from the Father of lights, with whom is no variableness, neither shadow of turning. 18Of his own will begat he us with the word of truth, that we should be a kind of firstfruits of his creatures.
James is saying that He conceived us "of his own will" and that everything good that we have came from him.
As Paul also said:
1 Corinthians 4:7 For who maketh thee to differ from another? and what hast thou that thou didst not receive? now if thou didst receive it, why dost thou glory, as if thou hadst not received it?
Will you answer that you did not receive faith?
Paul tells the Galatians:
22But the fruit of the Spirit is love, joy, peace, longsuffering, gentleness, goodness, faith, 23Meekness, temperance: against such there is no law.
Likewise, Paul tells the Romans:
Romans 12:3 For I say, through the grace given unto me, to every man that is among you, not to think of himself more highly than he ought to think; but to think soberly, according as God hath dealt to every man the measure of faith.
Romans 12:6 Having then gifts differing according to the grace that is given to us, whether prophecy, let us prophesy according to the proportion of faith;
But I have digressed. As I note above, the heart determine's man's choices, just like a tree's DNA determines its fruit. That's the Biblical analogy. A wicked heart brings forth bad fruit. And it's impossible for a wicked heart to stop being wicked on its own:
Jeremiah 13:23 Can the Ethiopian change his skin, or the leopard his spots? then may ye also do good, that are accustomed to do evil.
So, when you say "CAN" above, my answer is - their heart is so wicked, that it is impossible for them to do what is right. Do you call that God punishing them for having a wicked heart? I don't think so ... but why argue about that?
The point is that such people need salvation - they need a Savior. They need God to DO SOMETHING to rescue them.
David described it this way:
Psalm 40:2 He brought me up also out of an horrible pit, out of the miry clay, and set my feet upon a rock, and established my goings.
God is, as David said, our deliverer:
Psalm 40:17 But I am poor and needy; yet the Lord thinketh upon me: thou art my help and my deliverer; make no tarrying, O my God.
David also put it this way, speaking of the powerful Word of God:
7The law of the LORD is perfect, converting the soul: the testimony of the LORD is sure, making wise the simple. 8The statutes of the LORD are right, rejoicing the heart: the commandment of the LORD is pure, enlightening the eyes. 9The fear of the LORD is clean, enduring for ever: the judgments of the LORD are true and righteous altogether. 10More to be desired are they than gold, yea, than much fine gold: sweeter also than honey and the honeycomb. 11Moreover by them is thy servant warned: and in keeping of them there is great reward. 12Who can understand his errors? cleanse thou me from secret faults. 13Keep back thy servant also from presumptuous sins; let them not have dominion over me: then shall I be upright, and I shall be innocent from the great transgression. 14Let the words of my mouth, and the meditation of my heart, be acceptable in thy sight, O LORD, my strength, and my redeemer.
Man cannot change his own heart.
But again, I have digressed. The point is that man's choices, words, and acts, come from his heart. Man has a will that is free to choose what the heart desires. Thus, the heart determines the will. Depraved man has a depraved heart, which does not love God, neither can it. Paul explains:
Romans 8:7 Because the carnal mind is enmity against God: for it is not subject to the law of God, neither indeed can be.
Does that excuse the wicked heart or the carnal mind from responsibility simply because such a heart or mind cannot bring forth "good fruit" or "subjection to the law of God"? Of course it is not an excuse.
So, what do we have at the bottom line? We have the fact that man has a will, but that will is not random. It is purposeful and motivated. It acts in accordance with the heart. If the heart is evil, what comes out of the heart will be evil.
If man's heart were different could he choose to obey and to be subject to the law of God? Could he bring forth fruit? Of course - and man does when God converts that man.
That's why we pray for men's salvation: that God would change their heart. For that is what God does both to commoners and kings alike:
Proverbs 21:1 The king's heart is in the hand of the LORD, as the rivers of water: he turneth it whithersoever he will.
And, though man makes choices, God is behind the scenes arranging things according to His Sovereign purposes, for Scripture says:
Proverbs 16:9 A man's heart deviseth his way: but the LORD directeth his steps.
Jeremiah 10:23 O LORD, I know that the way of man is not in himself: it is not in man that walketh to direct his steps.
This is a difficult subject to study and understand.
Proverbs 20:5 Counsel in the heart of man is like deep water; but a man of understanding will draw it out.
I pray that God will give you, dear reader, wisdom to understand about how man's heart operates - and how God's counsel far exceeds man's. Pray for that wisdom, and if God will, you too will share in an understanding of the sovereignty and majesty of our Lord Jehovah!
"The righteous, and the wise, and their works, are in the hand of God: no man knoweth either love or hatred by all that is before them." Ecclesiastes 9:1
Praise be to our Merciful God!