> What about letting people opt-out of providing abortions? Or marrying gay couples? A case could made for people refusing to do these as matters of conscience as well.

They already can to both your examples. No doctor (excepting, perhaps, military doctors stationed overseas, or so I've been told but not confirmed) can be forced to perform an abortion. Similarly, no member of the clergy is forced to marry gay couples. The only people required to do so, now, would be state officials (justice of the peace) whose job it is to perform weddings and who work for the public (rather than being part of a private, though religious, practice).