The state medical board should pull his license, whether they will is an open question.
So because a person exercises their right of free association and intellectual property rights you want to go all big govt fascist and use lawfare to force a private citizen to render their labors and intellectual property against their will. In the 1800 that was known as slavery. People have the right to not preform labor for someone for any reason of their choosing that’s a God given right. Don’t like that this doctor is not an antivaxer move along to one that is then don’t be spoiled children running to mommy to force the other to play with you against thier will.
As to the legal aspects good luck with a suit only civil rights act or title IV protected attributes would be actionable via litigation. Health status is found in neither statue and is not a protected class nor attribute. Yes even HIV is not protected the sexual orientation of the person is though.