According to Royal Caribbean's website (
LINK)
"Passports are recommended for all US citizens for all travel outside of the United States. All guests are required to submit acceptable proof of identity.
Documents that will be accepted as proof of identity are: a passport; a federal, state, or a local government-issued picture ID card; an original or certified copy of your birth certificate with one of the following photo ID cards: state driver's license; US company employee ID card; or an ID card issued by a private identification card service. A voter's registration card is no longer acceptable on its own, but may be used as support.
Non-US citizens residing in the United States must show valid Alien Registration (green) cards prior to boarding the ship. Please check with a travel agent, government agencies, embassies or consulates to determine documentary requirements.
You are responsible for bringing all travel documents such as passports, visas, proof of citizenship, etc., required for boarding and reentry into the US. No refunds will be given to individuals who fail to bring the required documentation. For your protection, Royal Caribbean recommends that your passport expiration date not occur within six (6) months of the voyage termination date.."
So their reccomended, but not required. I have yet to go on one where i had a passport

Might as well just get one anyway just to be safe, its a good thing to have...