And what specific MORAL sense would that be? Don't kill? Don't steal? Don't deprive others of freedom, such as slavery?
Different religions, including christianity, have remained ambivalent or even encouraged these behaviors under certain situations over time. You're going to have a hard time arguing there is an immutable specific set of acceptable and unacceptable behaviors
First, don't mistake my point and leap to the next one. What I've looked at so far is just the EXISTANCE of a moral imperative, as distinct from a "this is better than that" instinct. Again, the difference between a "mistake" or a "bad idea" as opposed to a "morally wrong decision." So what I'm talking about is NOT whose moral code is right, but rather that the essentially universal existence of the moral impulse has implications all its own.
And here's where I'm going with that line of reasoning: While it's one thing to see people agree on social contracts, it's an entirely different thing when disagreements occur. And in our reaction to those disagreements we get a clue about the difference between "normative" and "moral" behavior. Consider the difference between the way we'd react in these situations: Someone comes to an important formal event having dyed their hair purple and wearing a Tux and tennis shoes(a violation of cultural norms); versus someone in the right clothing who then robs the host. In one case we feel variations on "that's inappropriate", but in the other we say "that's immoral." Or even more directly, think about civil disobedience of some law because it's immoral -- obviously, the protester is rejecting something his society has agreed on, so there's something BEYOND the society's standards that he's relying on.
Which leads to the more important point. Whenever people appeal to a set of standards almost by definition the standards are arising from a plane higher than the one they're operating on. Take a contract between two people. If someone violates the contract and refuses to return the other person's money, the contract doesn't enforce itself. Instead, the enforcement comes from a higher authority, the country's legal code. When a gang member's societal norms allow him to steal, the victim isn't going to think "Oh, what do you expect when cultural norms collide" -- He's going to feel WRONGED as well as angry. And in so doing, he's unconsciously calling on an assumed code of what's moral, independent of your culture and mine. That's the moral sense operating.
So to sum up for those who skip the text wall -- I'm not at all talking about whose morality is "right." I'm just trying to point out the presence of a moral imperative whose very existence suggests something about the nature of reality.