They can certainly invoke the First Amendment.
If the doctor knew the church’s stand against abortion when he took the position, it weakens his case against them.
Likewise the fact that he can no doubt find employment elsewhere as a doctor. It’s not like the market is saturated. The AMA sees to that.
Yeah, but getting a doctor fired if he only engages in legal practices and does not dip into the Medicaid till or rape patients with the AMA running the show is really difficult. I'm not excusing the bishop - I certainly give my own a certain amount of grief for not acting Catholic enough (heh heh heh), but I can see where they might be a bit timid.