It's ridiculous. I don't eat animals, but in my capacity at work I prepare and sell meat products. If I just suddenly refused to keep doing that, even always finding another person to do it, I would get fired. For once, I'd deserve it. It is incumbent upon me to go and find a job where I would no longer have to do that.
And if that doctor's other clients decide to fire her, that's perfectly fine. I fully support the use of private boycotts and sanctions where people feel it is appropriate.
Yes, because that was so incredibly successful in the defeat of Jim Crow and housing segregation outside of the Montgomery buses.
Simple math says you can deny service to 5%, 10%, 15% of the population with the support or non-interference of the majority and do just fine. Sucks to be in that minority, but it isn't usually white libertarian men who face total inability to buy a good or service, so we'll keep having these discussions until the end of time
If the couple were denied access to medical service for their child, you'd be raising a valid point, as I've already agreed earlier. But that isn't what happened. They weren't even delayed in having their child looked at that day. Granted, they were upset, angry, and dismayed. I would be too in their shoes. But as the Declaration of Independence points out, happiness isn't a right, only the pursuit of happiness.
Neither is there a right to be a doctor. Society can make rules that prohibit this sort of behavior. A doctor's license is a privilege, not a right. Do you disagree?
But then we're back to society forcing moral norms on a minority. Already we have forum posters arguing that doctors who won't perform abortions should lose their license, effectively barring a lot of Catholics and Evangelicals from the profession. In Canada, many law societies are refusing to recognize law degrees from faith oriented schools because of their stances on same sex marriage. Instead of the 1920's with "No Jews Allowed" such thinking gives the 2020's with "No Fundies Allowed".
While I find the doctor's action to be distasteful and poor theology, the couple found a doctor who will better serve them and the doctor doesn't have to violate their conscience. As shua noted this is just part of living in a pluralistic society, and we don't need to fix every injustice with punitive law.