Can ice make herniated disc pain worse?

No. Pain from a herniated disc is due to nerve root irritation. Ice is considered a pain reliever. Some nerve irritation can cause an abnormal hypersensitivity such that the cold is perceived as painful. It might be difficult for the person with herniated disc pain to differentiate that pain from the hypersensitivity associated with the cold because temperature and pain are transmitted on same nerves.
It may if there is. Muscle spasm present in the back or neck -occasionally patients find that ice may aggravate those symptoms if present from the disc herniation.