Yes, that patient can choose what, if any, information to make available. The patient has the right to choose who to make what information available to as well. I have the right to tell everyone that I broke my left forearm when I was 15, but that is certainly not all of my health information. I can also say that Aunt Sally cannot be told anything about me or even if I am a patient of the hospital or not.
Any information that the White House doctors have released is authorized by the patient and nobody else. So if his fever is released, it was released with consent or by breaking the law.
BINGO!