I believe this is the correct answer, it happened to me as well. I got friend suggestion for my daughter speech therapist, and all our communication and appointments were done only in paper form. But I searched her name on FB out of curiosity. :-)
OP: Can you exclude the possibility that the doctor just searched your name on FB? Why wouldnt he do that...?
> all our communication and appointments were done only in paper form
Just because your communication happened on paper doesn't mean that they haven't stored your email address or phone number in some service digitally, and that somehow that info has been shared (e.g. a phone with a facebook app that uploads all contacts)
A girl I talked to on Tinder a few years ago popped up on my People You May Know list. I befriended her and met up with her recently.
I brought it up to her, we wondered how it happened, and she admitted that she'd searched for my profile a week before.
I'd say this aspect of the system is a good feature. If you're searching for someone, it's nice to give them an opportunity to "independently" reach out to you.
OP: Can you exclude the possibility that the doctor just searched your name on FB? Why wouldnt he do that...?