My dad occasionally was called “Doc.” But he was a Pediatrician. But, for the most part, the people I meet who are nicknamed “Doc” are not the type to hold advanced degrees. Rather, they are usually more of the Harleys, tattoos, and big mustache variety. Where do people like this get the nickname “Doc”? (I did go to school with someone whose last name was Dougherty, shortened to “Doc”, but that was the only case like that of which I know.)
I’m guessing that, if not bespectacled, they are the person in the group considered to have the most ‘knowledge’ or ‘wisdom’ and so affectionately known as Doc. Could there be a link with a certain dwarf we all met as children ? His six mates called him Doc.
Just a wild guess.
Some Army Medics or Navy/Air Force Hospital Corpsmen get nicknamed Doc.
For obvious cause.
I imagine having a last name of “Holliday” almost ensures you of that monniker.
For example, Mike Holliday was an early member of Paul Revere and the Raiders. Can you guess what his nickname was? 
I once knew a guy whose first name was Doc.
IIRC “Doc” Sevrinson of Tonight Show fame was the son of a physician who let him follow him on housecalls as a little boy, earning him the nickname “Little Doc.”
For a while I was called Doc or Doctor by co-workers. One guy said it was because I looked smart. (?) The other said it was because I always had something to read. This was in a factory, where everybody was bespectacled for safety. I do hold a dubious Doctor of Divinity degree from the Universal Life Church, but neither guy knew that.
I’d always assumed that being handy with a knife was a good way to get a nickname like “Doc”, but I can’t back that up with a cite.
That’s “Mac”. 
(Now I’ve got Louis Armstrong stuck in my head. And he takes up a lot of room, damnit!)
Stranger
I actually know someone whose given name is Doc. And he’s Doc So-and-So IV, so his father, grandfather, and great-grandfather must be Doc, too.
My last name is similar to Hyde, as in Dr. Jekyll and Mr.
So in junior high school I had the nickname Doc. No one’s called me that since.
That would make sense if your name sounded like Jekyll, but since it didn’t, shouldn’t your nickname have been “Mis”?
Thinking like this is what got me beaten up in junior high school.
Remind me to smack you next time I see you.