What is the name 'Doc' short for?

Answer 1

As a proper name (rather than a title) "Doc" is actually of Welsh origin and not short for anything - it's just a short name! It derives from "Madoc" or "Maddock". Sometimes used as a nickname too, for example Doc Holiday was really called John but he was a dentist hence his nickname.

Answer 2

"Doc" is short hand for Doctor. Enlisted Army Combat Medics are often called "Doc." No, they are not real medical doctors, but they are the closest thing to a doctor in heated combat before medical evacuation. The medic must have won their collective confidence to be called "Doc." Ph.Ds. are often called Doc. Its meant to be a term of informal respect.