No one should be banned from a profession due to their beliefs. However, if these beliefs come to affect the way a person performs their professional duties, they must choose one of the other. They have no right for special accommodation and should not be held to a different standard as anybody else.
I already touched upon this in the other reply I did this morning in this thread, so this will be similar. The doctor here had been recommended to the couple, which suggests that she is one who seeks to establish a rapport beyond merely the technical aspects of medicine. While that is a good thing, it does mean that she won't be able to establish a rapport with everyone.
But what if there were more doctors who espoused this kind of behavior when dealing with LGBT patients? That means this patients would be forced to choose their doctor among a much smaller pool of physicians, and thus be at a significant disadvantage compared to other patients. If you think everyone should have access to the same quality of health care, you can't have doctors pick and choose their patients.
I acknowledge the potential problem. But suppose the situation were reversed? Suppose the doctor were a lesbian and the parents were bigoted fundamentalists who would each time they took their child in would wear anti-LGBT tee-shirts? Would you force the doctor in this situation to treat their child or you would you allow that doctor to tell them to find a more compatible one?
The law already would would it not? As being LGBT is not a protected class but religion is.
You have this sort of 'meh' reaction on the forum
every time issues like this come up when it comes to LGBT people and their families. I know your 'couch libertarianism' means that certain equality laws don't sit well with you but given that race, colour, national origin, age (mostly), pregnancy, citizenship, family status, disability status, veteran status are all protected classes, don't you consider that perhaps sexuality should be? I mean, can you be so gracious? Because you always seem to think we should just try the next door down every time it's slammed on our faces.