there are stories out there that he will be fired over it.
He was suppose to be a good little target that does not shoot back.
The doctor isn't necessarily a hospital employee. IIRC, the majority of psychiatrists are still self-employed and maintain private practices. They bill the patients and their insurance companies directly, and pay their own practice expenses. He might rent or lease office space from the hospital. That's fairly typical, particularly for specialists that need to spend a lot of time in the hospital. If he's not a hospital employee, he can't be fired by the hospital. If he is in violation of medical staff rules, he could possibly have his hospital privileges revoked. However, I don't know if how likely it is that the hospital staff rules cover firearms, and even if they do, it's other doctors, not hospital administrators that make the decisions about medical staff privileges, so that isn't likely to happen. If the hospital's lawyers insist, he might receive a letter reminding him about the hospital rule -- but since he clearly saved several lives by shooting the shooter, I doubt it.
I think this is mostly wishful thinking by the hoplophobes.