Been medically discharged from Marines DEP, can I get into a different branch?

I have been medically discharged from the Marines Delayed Entry Program over 5 months ago and I want to enlist into another branch. I heavily sprained my wrist during a workout (no broken bones) and was discharged. I was quite upset and confused. I have always seen myself committing to the military, but i have been told that I cannot reenlist in the Marine Corps. My question is: Could I enlist into another branch (Air Force, Army, etc.) despite my medical discharge?