The crew members’ cases come after two passengers sharing a room on the Celebrity Millennium, another Royal Caribbean ship, tested positive last week for Covid-19, though the cruise sailed on. Cruise operators are hoping to show they have protocols in place to handle Covid cases, in contrast to some high-profile outbreaks that brought sailings to a halt last year.

Six of the Odyssey crew members are asymptomatic, while the other two have mild symptoms, Mr. Bayley said. All crew members are quarantining for 14 days with routine testing, he said. The company will also delay a test voyage on the ship scheduled for late June.

All 1,400 crew members on the ship were vaccinated on June 4 and will be considered fully vaccinated on June 18, Mr. Bayley said. The cases were detected after the vaccination was given and before they were fully effective, he added.

“Two steps forward and one step back,” Mr. Bayley said. “While disappointing, this is the right decision for the health and well-being of our crew and guests.”

The U.S. Centers for Disease Control and Prevention defines the time frame for full vaccinations as two weeks after receiving the second dose of a two-shot vaccine or after a single-dose vaccine. The crew members haven’t gone past the 14-day mark after their vaccine shots, thus leading to a higher risk of disease spread, said Rebecca Lee Smith, an associate professor of epidemiology at the University of Illinois at Urbana-Champaign.

“One of the concerns with this group is that they weren’t fully vaccinated yet, and so they were still susceptible,” Dr. Smith said.

A Royal Caribbean spokesman said the cases don’t affect the company’s other sailings. Celebrity Edge, another of the company’s ships, is set to depart from Fort Lauderdale on June 26, breaking more than a year of sailing hiatus in the U.S.

The CDC on Wednesday lowered its assessment of the risk of cruise-ship travel to level three from level four for people who aren’t fully vaccinated. It said the chance of contracting the disease is high as the virus appears to spread more easily among people in close quarters on ships. But the agency has approved several ships to resume with new policies or recommendations in place such as passenger testing and capacity limits. To be able to sail, cruise operators either have to conduct test sailings or attest that 98% of crew members and 95% of passengers are fully vaccinated.

There have already been signs of tension in the companies’ dealings with local authorities. Florida, as well as Texas and Alabama, which also have cruise ports, passed laws barring businesses from requiring “vaccine passports,” or proof of vaccination. Royal Caribbean has shifted from requiring vaccinations to recommending them for coming Florida cruises, though it is still requiring them for Alaska voyages.

Write to Dave Sebastian at dave.sebastian@wsj.com