Kass says, We are now too sophisticated for such argumentation; we wouldn't be caught in public with a strong moral stance, never mind an absolutist one. We are all, or almost all, post-modernists now.
Realizing that it is intrinsically impossible to prove the correctness of any moral principle, and why that is, teaches us also that a logical or scientific proof for a moral principle cannot be expected, no matter how correct the principle is. A proof cannot be demanded. This makes us feel more free to take a strong moral stance, if we really feel its correctness.

Psychology professor Paul Bloom, author of the relatively recent book Just Babies: The Origins of Good and Evil, said in http://www.samharris.org/blog/item/the-roots-of-good-and-evil that while some moral values “are the product of culture and society” and “not in the genes,” “there also exist hardwired moral universals – moral principles that we all possess." Though I haven't yet read the book, I would expect Bloom to theorize that those moral universals evolved in us due to their evolutionary adaptive value.
Neo-Darwinism seems to be the overwhelming consensus among smart people who have thought about it for a long time, and moreover I have no reason to doubt it -- as far as it goes. But I don't think it explains the universal longing for transcendent experience. Because, how would the behavior trait of "ability to access transcendent experience" favor someone in the propagation of their genes?
Nevertheless, I do think that we all have inborn, probably hard-wired (built into our brains), moral universals. I think that by deference to them we find the peace of mind that can enable transcendent experience. Hence the morality and selflessness of those men and women thought historically to have attained deep transcendent experiences.
I have also run across the argument that we all (i.e., universally) do have near-identical senses that we experience as moral senses, but which are not inborn. They are really just unconscious commitments to a pleasure-oriented, self-interested social contract. Our parents and teachers have somehow instilled in us ideas which we now experience only as instincts, such as the idea that by risking myself to save another, this will help build a society more conducive to helping me maximize MY pleasure. For certain reasons I find this idea less plausible than the idea that morality is inborn.
But under either theory, our senses of right and wrong (moral intuitions) are pre-logical. However they originated objectively, to us subjectively they are a kind of qualia. If logic or any kind of discursive thought was involved in their origin, we aren't conscious of that logic. We just feel that we should try to save someone. Or we possess “the wisdom of repugnance.” Among such feelings are our moral absolutes without which moral relativism would be correct.
And yet in spite of an impressive universality and uniformity, some moral intuitions do differ importantly from person to person, never more at present than in relation to the abortion issue.
I feel that the intuitions existing at the deepest mental level in each person do not differ. But to reach those deep and correct moral intuitions requires more introspection than many people will undertake, and those correct intuitions are overlain by whatever neurotic emotional needs may have developed in each person in childhood.
If we accept this framework, then we can see how difficult it will be (ultimately) to search and find correct moral intuitions, moral absolutes, through logic, thought experiments and discursive argumentation. I think that by probing inside we can all eventually find the correct moral intuitions within us, but even then we will not know from what logic, if any, they derived. So we cannot hope to reconstruct that logic just by thinking about it. And logic as an abstract exercise (not attempting to reconstruct some logic that may have originally engendered those correct moral intuitions) is, like science, committed to dispassion. Just as science can tell us what is, but not what should be, logic cannot tell us what should be or that we should care about anything. The logical part of our minds cannot care, and the caring part of our minds, which can say "should" or "shouldn't," cannot use logic.
So logic, thought experiments and discursive argumentation can be catalysts. They can jog our minds or nudge our minds toward correct moral intuitions. They are very important in this way. But to finally pass through the door to the caring part of our minds where the correct "shoulds" and "shouldn'ts" reside, we have to leave logic behind.

That's a pretty good strategy to use on the laymen moral relativists. The more sophisticated moral relativists don't mind biting those bullets at all.
But I'm not convinced it's all that relevant to convincing them that abortion is wrong.
The fact of the matter is, whatever someone's meta-ethical view, they are going to have reasons for thinking killing ordinary people is wrong.
It doesn't really matter whether they are nihilists or relativists because we can still examine those reasons and point out various inconsistencies such as the reasons justifying infanticide (assuming they don't bite the bullet).
But even if they do bite the various bullets it's entirely possible to argue them to a stalemate allowing them to come to an understanding of how reasonable the pro-life view can be. That might shake them up a bit and cause them to reflect a bit more.
I notice pro-choicers like to throw that meta-ethical red herring out there. I'm not so sure it's prudent to latch onto it. It doesn't seem to be all that relevant.
It'd be like two scientists arguing over which theory best explains some natural phenomenon and one of them going "well I'm an anti-realist so therefore I win." That's just supremely irrational.

That conversation is hilarious. It shows the relf-refuting nature of relativism. To say there is no truth is a statement of truth. Morality is slightly different than truth because of the different moral frameworks but much of the same reasoning applies to both.