The objective vs. subjective morality debate is really the theist vs. atheist debate in disguise since, with God, there is objective morality and, without God, there is no objective morality.
That's false.
Is it possible to clearly distinguish between objective and subjective morality? Can the moral status of particular actions (or classes of actions) be established independently of our subjective moral feelings and intuitions about rightness or wrongness? How would one go about recognizing the rightness and wrongness of actions apart from how we feel about those actions (or in the case of the utilitarians, the actions' results)? How might somebody be convinced that his/her moral intuition is in fact mistaken (if that makes sense)? What other evidence of rightness and wrongness is there?
Without getting into that debate, what evidence is there that human beings are bound by an objective morality and not just natural drives?
Human beings tend to broadly agree on the moralities that they accept. What's more, human groups in distant parts of the world share many moral principles in common. The "golden rule" (the principle of reciprocity) is an obvious example -- it's arisen independently all over the place. But there are some pretty dramatic divergences as well, such as differing views of slavery and of the place of women in the community.
Remember, a relativist would say that widespread moral unity among cultures across time is not enough since, at most, that would only be evidence of a BELIEF in objective morality?
Is there really a difference between objective morality and belief in objective morality?
My own approach to these questions is naturalistic. I tend to derive human morality from human social instincts. As a species, we evolved and are consequently optimized to live in social groups. That probably involves a propensity to favor kinds of behavior that strengthen group solidarity and effectiveness.
But at the same time, human beings are acquisitive self-maximizers, who are simultaneously driven to improve their own personal position, even if it's at the expense of their fellows.
So conflicts are going to arise, both internally/subjectively within individuals (conscience vs desire) and socially within groups (law enforcement). In human social groups, the sometimes self-abnegating social values are probably going to be embodied in real or mythological authority figures who represent the group and its interests -- clan patriarchs, mythical progenitors, totems, gods and kings.
Put another way, I don't think that solitary animals, animals who live alone and don't have to accomodate to others of their kind, have anything closely approximating morality. Social insects like ants on the other hand are probably just about 100% morality (group interest) with very little self-interest. Herd animals, and even more obviously pack hunters like wolves, are probably a lot like us humans, with some non-verbal and un-mythologized analogue of a conscience. They have personal desires but can sense when their individual actions cross the social line, and feel something approximating guilt and shame as a result.