Depends on how you hurt it. I'm no doctor or physical therapist, but I'd say that there are definitely ways a bad golf swing could hurt your wrist. I've done it before by hitting too much ground and "jamming" or spraining my right wrist.
I changed my grip recently and I too have some feeling of pain in my left wrist. I'm not certain if it is because I'm doing something wrong or if it is just a result of weak muscles being worked as I played three 18-rounds during the weekend.