1. If God does not exist, then objective moral values do not exist
Support: If there is no transcendent anchor for moral values, then there is nothing morally binding and all things are technically permitted, even though some behavior may have become socially disadvantageous. I.e., humans are just one more animal on the evolutionary tree, and thus like primates may have learned to live together in a society, but our actions carry no moral weight.
Objection: Morality is rooted in whatever leads to human flourishing.
Response: Why? Why human flourishing specifically? Why would inflicting harm on another person be wrong? Animals rape all the time, and it's just nature at work. According to atheism, humans are just one of the animal species, so why treat humans differently? Why put humans up on a pedestal like that?
Objection: Moral values supervene on certain natural states (that is to say, moral values are brought about by the sum of several natural states, which values they themselves do not posses; i.e. moral values are a sum of the parts).
Response: If atheism is true there is no reason why moral badness would supervene on, say, a man robbing a store. A full accounting of all the natural processes involved in the man robbing the store does not seem to bring about the moral value of badness. This is the old is/ought problem highlighted by David Hume. The facts of the matter (the “is”) do not seem to bring about a moral obligation (the “ought”).
Objection: If morals are objective truths, then they are necessarily true and thus have no explanation for their existence (like numbers).
Response: Necessary truths do have explanations. For instance, "2+3=5" is necessarily true because the Peano axioms are true. Or the proposition "no event precedes itself" is necessarily true because "temporal becoming is an essential and objective feature of time" is necessarily true.
Objection: Objective moral values are anchored in Platonism (Platonism holds that abstract objects, such as "redness," actually exist apart from their use as attributes for concrete objects).
Response: This makes no sense. What does it mean to say that justice just exists, even if no people were around? And if it does just exist, why align your life with any particular one of these moral objects? Where is it written that one is good and one is bad? And third, it seems implausible that creatures would evolve that just happen to conform to these values as if the values "knew" we were coming.
2. Objective moral values and duties do exist
Support: In our moral experience we do perceive objective morality. A man who rapes and murders a child is not just expressing a socially disadvantageous behavior, but is just as wrong as the man who says that 2+2=5. Over the course of history, we witness what we might call "moral progress." Pedophilia and gladiatorial combat used to be accepted in ancient Rome. Slavery was once accepted in America. Jim Crow laws were once accepted. This has the appearance of progress, not just change. If morals were relative, then we would think of these as change but not progress and would accept that we might return to Jim Crow laws, slavery, etc at some point. I.e., we are slowly uncovering an objective moral "realm" just as we are slowly uncovering the physical realm.
Objection: Objective moral values do not exist because they are just tools that evolved.
Response This commits the genetic fallacy of saying that explaining how a belief originates invalidates the truth of the belief. We evolved eyes slowly through evolution, but this does not answer whether the objective external world exists or not.
Objection: There is no justification to believe that objective moral values exist because they are just tools that evolved.
Response: This presupposes atheism, the very position that is in question, and thus leads to circular argumentation.
3. Therefore, God exists
Support: From 1 and 2.
Objection: The Euthyphro dilemma. Either God commands something because it is good (and hence objective moral values exist apart from God), or something is good because God commands it (and hence what is good becomes arbitrary; on God's whim).
Response: This is a false dilemma and there is a third alternative. Morals are grounded in God's nature, not in his arbitrary whim. It is internal to God, and thus his commands only express his internal nature. "Whatever a good God commands will always be good."