The Grandeur of the Seas ship made it to Freeport, Bahamas, on its own power, but was escorted by U.S. Coast Guard vessels, CNN reported.
The fire was 2:50 a.m. in on the aft mooring deck and spread the crew lounge area on the fourth deck before it was extinguished just before 5 a.m., officials said.
The cause of the fire was not reported.
There were no injuries due to the fire, but two people fainted and another person sprained their ankle, Royal Caribbean said.
Spokeswoman Cynthia Martinez said there was no other information available on the medical reports, "but the ship has told us that there are no medical emergencies onboard."
The Grandeur of the Seas makes a seven-day journey that includes stops in Port Canaveral, Fla., Nassau, Bahamas, and CocoCay. It launches from the Port of Baltimore in Maryland.