In his most recent Q&A, Dr. William Lane Craig stresses the importance of distinguishing between moral ontology (the objective status of moral facts or properties) and moral epistemology (how we come to know moral facts) when deploying the moral argument for the existence of God:
I’m convinced that keeping the distinction between moral epistemology and moral ontology clear is the most important task in formulating and defending a moral argument for God’s existence of the type I defend. A proponent of that argument will agree quite readily (and even insist) that we do not need to know or even believe that God exists in order to discern objective moral values or to recognize our moral duties. Affirming the ontological foundations of objective moral values and duties in God similarly says nothing about how we come to know those values and duties. The theist can be genuinely open to whatever epistemological theories his secular counterpart proposes for how we come to know objective values and duties.Apologists have been letting atheists get away with ignoring this distinction for too long, and sadly they themselves sometimes ignore it, when they imply that it is belief in God that is essential for people to act morally, when the heart of the issue is whether objective moral values or properties exist, and what is the best explanation for the existence of objective moral values or properties. In fact, we need to keep the following questions separate when we talk about moral theory (this is a tripartite division I came up with for my ethics class):
1) What is the content of morality, i.e. which actions are right and wrong, and how do we know it?
2) Why should we be moral?
3) What is the ontological status of moral statements?
It is a big mistake when moral arguments center on question 2 or even question 1 rather than question 3. These are all important questions, and of course answers to any of them will have implications for the other two (how we come down on the status of moral statements will obviously have implications for our motivation to be moral), but the only relevant one when it comes to the moral argument is question 3.
Notice how many objections become irrelevant when we maintain a proper focus. As Craig notes, it is no objection to the argument that unbelievers can know their moral duty and act upon it. It is no objection to the argument that there is disagreement about the content of morality, because the moral argument only hinges on the acknowledgment THAT there are objective moral facts. It does not hinge on the acknowledgement of any specific moral facts, although getting people to admit that, yes, they know murdering innocent people is wrong, not just distasteful, can help build the case for the existence of objective moral facts in general.