The moral argument for God's existence finds one of its most historically influential treatments in Dostoyevsky's magisterial The Brothers Karamazov. Consider the following passage where Smerdyakov replies to Ivan (the atheist):
I first thought that if I had some money I could start all over again, either in Moscow or, better still, abroad; I got that idea, sir, mainly from 'everything is permitted'--it was you who taught me that, sir, because you used to say it a lot--because, if there is no eternal God, then there is no virtue and, what's more, absolutely no need for it.
Christian apologists have frequently adopted Dostoyevsky's treatment of the argument. For instance, Peter Kreeft argues: "Is it not true, as both Dostoyevsky the Christian and Sartre the atheist say, that 'if there is no God, then everything is permissible'?" (Three Approaches to Abortion, 50).
The unstated premise in such arguments seems to be that everything (e.g. every free human action) is not permissible. Thus the fact that atheism permits everything counts against the truth of atheism. So what is to be said for this argument?
If we are to assess the worth of the argument we have to distinguish two distinct arguments. Consider again Smerdyakov's closing sentence: "if there is no eternal God, then there is no virtue and, what's more, absolutely no need for it." Thus, the existence of God is tied to (1) the existence of virtue and (2) the obligation to satisfy virtue.
We can call the first argument the "Virtue existence argument." According to this argument, the existence of objective virtue and vice or value and disvalue depends on the existence of God.
The second argument is the "Virtue reward/punishment argument." According to this argument the existence of objective reward and punishment, or the satisfaction of justice, depends on the existence of God. If there is no God then ultimately there is no one to lower the boom if we fail to pursue virtue in our moral lives. (Those familiar with Kant will see here an inchoate version of his argument for the existence of God from practical reason.)
The virtue reward/punishment argument seems to me certainly correct in one sense. If there is no human being more powerful than Genghis Khan to call him to account, then anything is permitted for Khan. But the argument seems weak in another sense since it is very doubtful that our strong inclination to see the standards of justice satisfied (that is, to see Khan get his comeuppance) provides in itself a very persuasive argument for God's existence. Maybe this is just the kind of universe in which virtue, though existent, is not necessarily rewarded.
The more promising argument is the first. As the virtue existence argument goes, the existence of objective virtue or value could only exist if God exists. Thus, if certain free human actions exemplify objective moral value or disvalue, then this implies the existence of God as the source of that value.
Unfortunately I don't think the virtue existence argument is particularly persuasive. But that is not to say that it is a complete misfire. It certainly does refute naturalism which claims that only the spatio-temporal continuum that is the object of scientific enquiry exists. On the value view there is at least one dimension to reality irreducible to the spatio-temporal and that is the moral universe in which free actions have moral value or disvalue.
But the fact that the existence of objective value counts against naturalism does not mean that it necessarily counts for God. Rather it only counts for supernaturalism, and that is compatible both with objective value being rooted in the necessary willing of a personal agent (i.e. God) or with it being self-existent as in Plato's form of the good.
So to sum up, we have here an argument against naturalism, but it underdetermines whether personal (i.e. theistic) supernaturalism or impersonal (i.e. platonic) supernaturalism is the best explanation of moral valuation.