Per the headline, the doctor isn't refusing questions on doctor/patient privacy grounds, but on 5th Amendment grounds, meaning, the doctor doesn't want to say anything that would incriminate him in a crime. Pretty damning, at this point, as to his knowledge that he may be complicit in a criminal cover-up.
It shouldn't. Regardless of the employer, if the doctor is seeing the patient, the patient is the one who has the privacy rights. Think: military doctors. Obviously paid by government, but their patients still have protections.
There are exceptions that can be made to those privacy rights, but until those exceptions are met, the patient still has those rights, it isn't the doctor's decision what to share.