I. The argument from Implied Practice
1. If ethics is subjective, then we should expect people to recognize that actions which they are inclined to think of as "wrong" are only wrong from their point of view.
2. But invariably, people view wrongs against themselves as actions that are really wrong.
3. Therefore moral values are objective and not subjective.
This is actually an argument against relativism, not against subjectivism. The issue is whether morality depends on one's point of view - is morality the same for everyone? I will interpret this argument with "subjective" replaced by "relative" and "objective" replaced by "absolute."
But if morality is relative to the social or cultural group, rather than to the individual, this argument doesn't work. The people with whom I interact on a regular basis are, by definition, part of my social group. If morality is just a group's implicitly agreed-upon restrictions on behavior, then of course I will expect those restrictions to apply to others in my group, and I will feel justified in being upset when those restrictions are violated.
What about the case where I feel wronged by someone from a different culture? The absolutist might claim that this proves morality is not relative to the group.
But the relativist can argue that this response is a result of how ingrained our sense of morality is. I will tend to assume that anyone with whom I come in contact is part of my social group and subject to the rules I am familiar with.
Furthermore, there are cases where we do make allowances for cultural differences. I might feel offended if someone cuts in line in front of me, but if I learn they come from a country where pushing to the front is standard operating procedure, I might think, "Oh, that's OK, they didn't know how we do it here." Or imagine that someone steals from me, and I later learn they come from a culture where all property is shared communally. Then I might not feel a sense of moral outrage at the act - though I would probably still want my property back.
Finally, there is no problem here under a non-cognitivist account of morality, either. For the non-cognitivist, moral disapproval amounts to saying, "I don't like what you did," or "Don't do that!" or both. When someone does something that hurts me, these would be very natural reactions, not requiring any sort of absolutism.