Using the word "doc"
I'm examining and touching on your private parts and information, I'm ok with you calling me "Doc". I'm a retired military physician (MD). Concerning military medics (enlisteds) in the field, they are pretty well trained, because they want to be and they have to be. We work hard to give them all the medical training possible. If you get hit or sick out there, the guy treating you is doctoring you. He IS your doctor. England, Australia, Japan, etc, have 6 year medical schools to which high school graduates matriculate without doing a 4 year undergrad program . Also their degrees are bachelor degrees, not "doctors of medicine', but still they are just as good at doctoring as we are over here. "Doc" is neutral, and actually I prefer "Doc" , both in the office and out. No need to call me "Doctor of Medicine" or even "Doctor" . "Doc" differentiates me from others in the room, and refers to what I am here for. To me, the rest of the word or title is superfluous in conversation.
It depends on the relationship. You could call a president "Prez", or a professor "Prof" – either of those would be acceptable English, but whether or not you'd be violating social etiquette is another matter. Some people in authority have no problem with a more informal title, while others might be uncomfortable with it.
Calling your health care provider Doc is like taking a new medication or starting a new exercise regimen: you should consult your doctor first.