The cruise, ironically designed to prove that sailings could go ahead safely in the US, was scheduled to depart in late June but has now been postponed, with no replacement date confirmed.
The positive tests have also impacted the inaugural sailing of the line’s new Odyssey of the Seas ship.
Due to set sail from Fort Lauderdale, Florida, on 3 July, the liner will now make its maiden voyage on 31 July instead.
The news was shared on the brand’s Facebook page by Royal Caribbean CEO Michael Bayley.
“Two steps forward and one step back!” he wrote.
“Guests and travel partners will be notified and given several options to consider.
“While disappointing, this is the right decision for the health and well-being of our crew and guests.”
The cases were picked up in a routine test of all 1,400 of the Odyssey of the Seas’ staff.
Of the eight employees who tested positive, six are asymptomatic.
All crew are now quarantining for 14 days “to protect the remaining crew and prevent any further cases.”
In May, Royal Caribbean was the first US cruise line to receive the go-ahead to run “simulated voyages” from the Centers for Disease Control and Prevention (CDC).
These tests cruises, filled with volunteer passengers rather than paying customers, are mandatory for cruise lines in the US ahead of restarting commercial operations to ensure that Covid-safe protocols are in place and fully functional.
However, the requirement is not necessary where 98 per cent of crew and 95 per cent of passengers are fully vaccinated.
As of 18 June, all crew aboard the Odyssey of the Seas have been jabbed.
Royal Caribbean caused a stir when it first advertised for volunteers for its cruises back in November 2020.
Some 100,000 people signed up for the company’s “mock cruise” programme within five days of details being shared online.