Can i get arthritis from a wrist injury?

I hurt my wrist snowboarding, The pain wasn't sever enough to where i felt it was necessary to see a doctor, I figured it was just a sprain. Its been a year now and my wrist hurts every once in a while, Like when i do push ups, Curling weights, Turning something that requires alot of strength etc. Could i have developed arthritis? Or just a damaged tendon? or am i being paranoid lol im 22 years old. Im also planning on joining the military soon so im afraid if i get checked out by a doctor there not going to let me join.