I see no reason for him to resign without knowing the nature of the postings. Your suspicions are probably right.
I know parents who have keylogged this own kids and examined their phones and computers to know what they are up do - it’s a parents right. Kids ought to act like someone “is” watching...
There have always been teachers who act inappropriately, what is new is that it’s no longer predominantly just the males.
This is just one reason I deactivated my facebook.
That, and I don’t need all my collegiate indiscretions available to future employers.