Trump is entitled to the same HIPAA privacy rules as a private citizen is. Dr. Jackson won't give out any information that his patient doesn't want released. And I highly doubt that mental health was part of the physical, nor should it be.
It has been part of every physical I ever had with a flight surgeon and with my private physician.
The evaluation is made by observation and rudimentary questioning something as simple as "How do you feel?", "What do you do with your spare time like hobbies?" and "How are you sleeping and eating?"
The Dr also observes your mannerisms like fidgeting, twitches and tics. They may ask you to do simple tasks that you don't even think about from a psychological point of view. Something like "Stand and turn to your right and look straight ahead" while they shine a light across your pupil. They are testing cognition, reflexes as well as look for signs of psycho-neurological abnormalities.
These may not seem like probing questions or actions but the answers/reactions may indicate underlying problems that would indicate a referral to a psychiatric specialist.