Until the AMA determines positively there is a link between gun use or ownership and some physical or mental disease, they have no valid reason to question anyone about gun ownership.
If the doctor believes someone is "dangerous," they need to contact law enforcement. HPD already knows if you possess a gun legally. If you have a gun illegally, I doubt you're going to tell the doctor that!
Bottom line, personal information should never be shared without knowing the reason and whether it's being stored or shared elsewhere. Without a valid reason to ask, the doctor should keep his questions to himself. If he's asking on behalf of his employer or any government agency, they should disclose that up front before asking.
I don't tell my gun dealer all my medical issues, nor do I tell my doctor what guns I own. "Need to know" must be paramount.
BTW, the doctor doesn't have 1st amendment protection for asking questions of a private individual. That's ridiculous! I suppose Cops also have a first amendment right to question a private citizen not detained for any reason, but just to find out where he's going, where he's been, what's in his pockets, who does he hang out with. Seems reasonable. Since the Cop has to see the results of crime, he should be able to use questions to obtain probable cause.