Congress could get ahold of his medical records, but there are, of course, valid privacy (HIPAA) issues. Those apparently don't apply in getting evidence for an impeachment trial. I also (don't specifically know) if the request would go to Walter Reed - where the doctors who examined him are - rather than, say, an official in the administration, per se.

And the doctors at Walter Reed, perhaps being career medical staff and not political appointees in the administration, would prefer to put country first.

If this scenario were to happen, try to imagine the blowback for all the cabinet, etc. who would now be implicated in the cover-up.