De Waal’s argument is that moral impulses exist in our non-human animal relatives–particularly our closest relatives, the primates–and that we can see morality emerging along a continuum as a completely “natural” phenomenon. I think posing the issue as “morals (with or) without God” is misleading, though. I don’t see why a theist (Christian or otherwise) should have a problem with the idea that morality is “natural.”
On the contrary, it makes good sense to me, theologically speaking, to say that the potential for morailty would be natural to human beings. God–so the Christian tradition says–wants us to live lives characterized by mutal care and support. This kind of life-in-community is what we were made for.* This is directly contrary to the idea that morality is some kind of divine “add-on” to the human condition. I see de Waal’s proposals as something like an updating of Aristotelian-Thomist ethics in light of current knowledge. Morality is basically about those rules, practices, character traits, and so on that are conducive to human flourishing.
The other question that comes up is that of sanctions. Will people be moral if they aren’t afraid God will punish them? I think that, one, it’s just an observable fact that a lot of people behave perfectly decently without worrying about divine punishment (including many religious people), and, two, basing morality on the fear of divine punishment is bad theology. Paul’s letters in particular make it clear that a Christian’s motive for morality should come from a freely given response to the love God has shown us, a response empowered by the free gift of God’s Spirit.
Where we might see a distinctive human quality here (and I realize that arguing for “unique” human capabilities is risky) is in our capacity to take a “God’s-eye” perspective on morality. What I mean by that is that we can grasp a moral vision that goes beyond our immediate kin group, beyond our friends, beyond our religious and national ties, even beyond our species, to encompass all God’s creatures. An example is the imaginative depictions of God’s eschatological peace in the Hebrew prophets. This is consistent with some recent theology, which has located the “divine image” precisely in our function as God’s stewards of creation. On this view, our “dominion” is properly aimed at facilitating the widest possible flourishing of God’s creation.
*Talking about “what we’re made for” obviously means something different in an evolutionary picutre of the world than one in which humans are understood to be a direct, special creation of God, but I think we can say that God intended to bring about creatures like us through the evolutionary process, in addition to whatever other divine goals that process serves.
UPDATE: In retrospect, “Monkeys, Morals, and God” seems like the obviously superior title for this post. I’m really terrible at coming up with catchy blog post titles.