I know this is going to sound very libertarian of me, but it's not as if this doctor refused to provide emergency care. She even made arrangements so that the baby would still be seen by a doctor at the appointed time. The idea that people who choose to provide services should have no ability to decide who to serve is a rather illiberal one. It was a bigoted and stupid decision on her part, but unless it rises to the point of preventing people from having any access to a needed service, I don't think government should be interfering in this particular form of idiocy.
Yeah, I agree. She's clearly an awful person, but at least she showed the basic human decency to make sure the baby was cared for by someone who wasn't a bigot. I don't think it should be actionable.
What if another doctor hadn't been available. Her religious "beliefs" still would have dictated that she not treat the baby.
If she let her religious obligations keep her from treating a baby when no one else was available, that would violate the Hippocratic Oath and she should lose her license. It's hypothetical, though.
The fact that she arranged another doctor is the deciding factor for me. She didn't abandon her responsibilities, she passed them on.