My money's on Dr. Greene. Cooperation is a feature of game theory; it's how plus-sum games are won by everyone. Most cooperative action needs to be coordinated by authority. Law is one source of authority, but morality is another. Internalizing that coordinating authority is complex, but a combination of genetic wiring and epigenetic social influences combine to make it happen. We call our cooperative plus-sum behavior - when we cannot justify it by economics - morality.

Whether morality is "universal" isn't worth asking. There are some moral behaviors that seem to work everywhere and all the time. Others, are more local, more contingent on abundance and scarcity, technology, etc. The "universal" part is the cooperative effort mediated by the authority of mores.