Liberalism and moral relativism
I think what we're trying to do is explain an odd phenonomenon, namely the tendency of conservatives believe that liberals are moral relativists. It is odd that some conservatives believe both that liberals are moral relativists and that they are scheming to foist their gay/vegetarian/pacifist/socialist moral agenda on everyone else.
There are several explanations for the perceived link between liberalism and relativism. Matt already talked about the observed correlation between secularism and relativism. Conservatives may have observed that people who abandon religion sometimes become relativists instead of adopting a secular moral theory. Alternatively, if they believe that their religion is the source of all morality, and that liberals are overwhelmingly secular or ecumenical, then they may conclude that liberals lack absolute moral standards and must therefore be relativists.
I think there's a deeper explanation for the confusion, though. Liberal ethics is sometimes confused with relativism because the two theories give the same advice in some cases, albeit for very different reasons. The conservative bogeyman relativist is someone who thinks that everyone creates their own "moral reality" and that nobody can judge anybody. When speaking casually, liberals sometimes say things that could be uncharitably construed as relativistic. We say things like "Who are you to judge what consenting adults do in the privacy of their own home?" or "How dare those religious fanatics impose their morality on me?" These statements aren't really relativistic because they're backed by unarticulated premises. When liberals don't accept the argument for the moral superiority of one option over another, they counsel people to choose whichever one they like. This, to conservatives, may sound like relativism.
Let's say a conservative is arguing that gay sex is wrong. As a non-relativist liberal, I dispute that claim. I'm not taking a relativist position that gay sex is wrong for him, but right for other people. Instead, I'm claiming that the conservative has his moral facts wrong. In this non-relativistic vein, I will offer arguments to to show that consensual gay sex and consensual straight sex are morally equivalent. If neither option is morally preferable, it follows that individuals should be free to make whatever choices they want. Superficiallly, this position might look like relativism because it offers no moral guidance in choosing what kind of sex to have.
Likewise with abortion. Liberals argue that a woman has the right to decide to terminate her pregnancy. The standard line is that the choice between early abortion and gestation is a choice between morally equivalent alternatives. Unlike relativists, liberals maintain the moral equivalence holds regardless of the prevailing cultural norms or the preferences of the individual.
Pro-lifers are taking the position that gestation is right and abortion is wrong. Pro-choice liberals argue non-relativistically that both abortion and gestation are morally permissible, and further, that that the pregnant woman has the right to make the final decision about her pregnancy. A relativist might say that a mother's desires determine whether an abortion is the right thing for her to do. Liberals argue that a mother has an objective right to choose (even if she doesn't believe it), and that she has this right in virtue of the non-relative fact that abortion is morally equivalent to gestation.