The type of physician on board varies from ship to ship. All are M.D.s or D.O.s and ACLF (advanced cardiac life support) certified. Chances are they will not be surgeons although I have known a retired surgeon who worked as a ship physician. But even if the physician is a surgeon they would not do surgery onboard unless there was no possible alternative. While the infirmaries are equipped on par with U.S. E.R.s (I've been in several ship infirmaries, so I know of which I speak) that does not mean they operate when other options are available. Should your son have a hot appendix they would off-load him at or to the nearest port. Whether they would treat him locally or fly him to the States would depend on his condition and your locale.
I would have a VERY serious discussion with his doctor about the situation. A ruined vacation is nothing in comparison to the health of your child. You have the insurance - use it! Reschedule the trip for his next school vacation.