I might be mistaken, but isn't the agency that grants/revokes liscenses not a government entity, thus Free Speech is a non-issue. On an Ethics level, he creates a hostile environment that seeks to exclude care to people or create the illusion that certain people would get sub-standard care. I believe that is very much against the code a doctor swears to in order to get thier ability to practice medicine.
At the very least, an Obama supporter would have a much easier time getting a judgement for medical malpractice if anything went wrong.
To me, it's the same as a doctor posting a sign on his door that if you're black, go somewhere else for treatment.
Doctors are licensed by individual states, not private political organizations (aka AMA). So free speech does apply.
As for you other link, do US doctors actually use that oath? or do they use the Hippocratic Oath?