The big problem with this, at least on the anti-abortion side, is that in their minds many of them feel that they are protecting innocent lives, saving babies, and these doctors are murdering actual people.(*) Whereas if somebody is homosexual, they're only sending themselves to damnation unless they manage to corrupt some other innocent and lead them into a life of sin as well. And so the idea that it's not their place isn't going to play well.

(*)I do notice that even the rabid ones who want to charge doctors who perform abortions with murder never mention charging the mother, who requested this medical service, with the same charges. I don't know why this is.