Bottom line is medical records belong to the patient. The method of acquiring them may be problematic, though. It seems reasonable that the president can simply take them.
@skeeter You and I can get copies of our records but we can't take our original records. That would be considered theft and a violation of HIPAA laws. The doctor would have to report it to the Office of Civil Rights. The original record belongs to the provider (aka doctor or hospital) and they are legally required to maintain it for up to 30 years, depending on about 60 different variables. If the doctor quits his/her practice then they typically turn their records over to a hospital or another doctor.
The President has different rules. National security an all because they contain information which could be used against him. For example if the President was allergic to peanuts, well that would be in his records. Someone getting those records could then plot an assassination using peanut oil.
I'm sure this exact same thing has been done with every single President for the last 60 years.