Can I join the navy with a herniated disk?

No. No. "ruptured nucleus pulposus, herniation of intervertebral disk or history of operation for this condition" are all causes for rejection for induction, enlistment or commissioning into all branches of the armed forces.
Don't know. If you have symptoms as a result the cervical disc herniation such as arm pain, numbness and tingling or associated leg weakness and gait disturbance; i would recommend that you correct the condition prior to enlistment. If you are not symptomatic, then i can see no contraindications. Suggest you check with the appropriate recruiting offices for the army, navy, airforce & marines.