Speaking for myself, I am a prior U.S. Marine, but I inter-service transferred into the Navy to become a Corpsman, (and subsequently retired from the Navy)
Your father was every bit a Marine (more so for his service during Korea) but in reality a FMF (Fleet Marine Force) Corpsman.
The greatest honor for a Corpsman is to be called "Doc" by his Marines.
But as previously suggested find his dd214, that will settle the question.....
Semper Fi!
Thanks again to all who have responded so kindly. Your expertise has been very helpful.