I'm not the OP, but doing this right now and I can say that no, you do not provide a doctor's note. It also does not have to be (at least in the US) anything actually affiliated with a specific religion. Religion is defined as a set of moral values you as a person live your life by. So if you were a vegan and did not want to eat meat, you could claim a religious exemption even if you were not a recognized religion. You could get a religious exemption as an Athiest, Agnostic, etc. You simply need to let the cruise company know that taking a COVID vaccine goes against your religious beliefs and then you'll likely need to list why, to keep the cruise lines in compliance with the CDC.