On its face, the argument against relativism is somewhat compelling. However, what is rarely mentioned is that what is being defended is something called absolutism, where experience is judged by and made to fit the abstract, absolute authority of a source that is often supernatural, or, as a sympathizer would frame it, "divine" in nature. Homosexuality is decreed by a sacred text to be a sin? No problem, pretend someone isn't gay or send them to a "conversion therapy" program. Theologically, dogma is the absolutists' enforcer.
To illustrate, William James and others have cited the Greek myth of Procrustes' "magic" bed. The Mythweb site relates the story like this:
Procrustes, whose name means "he who stretches", was arguably the most interesting of Theseus's challenges on the way to becoming a hero. He kept a house by the side of the road where he offered hospitality to passing strangers, who were invited in for a pleasant meal and a night's rest in his very special bed. Procrustes described it as having the unique property that its length exactly matched whomsoever lay down upon it. What Procrustes didn't volunteer was the method by which this "one-size-fits-all" was achieved, namely as soon as the guest lay down Procrustes went to work upon him, stretching him on the rack if he was too short for the bed and chopping off his legs if he was too long. Theseus turned the tables on Procrustes, fatally adjusting him to fit his own bed.UPDATE 9-18
One might reasonably hold any number of absolutist principles. For example, we all agree on the prohibition of slavery. We don't need an appeal to outside authority for that, but rather to our shared humanity. The abortion debate is so difficult because absolutist positions on both sides make some sense.