No, but if the government refuses to prosecute it or otherwise deter such actions, it’s most definitely a fourteenth amendment issue.
As far as doctors being fired for espousing unpopular opinions: I happen to support, in theory, the right of employers to fire employees who are undercutting their business’s core mission or values, even if all the employee is doing is creating bad publicity and even if it’s being done via speech, or religion, or politics, or “moral lifestyle choices”. But that’s not the world that the progresives want to play in, or the world that had been created under our current statutory regimen. So we may as well make ‘em live up to their own principles.