Lorry driver Mr Steele, from Tettenhall, was moved to hospital on Friday while his wife Wendy remained on board the ship.
The liner had been isolated in the port of Yokohama, with 61 people having been taken to hospitals after testing positive for the virus.
Wendy Marshall Steele has been in telephone contact with her husband. She said on Saturday that he was still feeling healthy.
“Alan is well, Japanese doctors are excellent,” she posted on Facebook.
“He is in a little room. Just Dr and nurse visit him.”
Mrs Marshall Steele said she was also well but had “cabin fever” as she continued to be confined to her room on the ship, a status shared with nearly 3,700 other passengers and crew.
“It was a hard first night without Alan. But hey, we are constantly in touch. Japanese doctors are excellent and he is in safe hands,” she wrote.
A total of 78 British passport holders – including crew – were among those who boarded the ship, sources told the PA news agency.
The ship’s operator, Princess Cruises, said the vessel’s quarantine was due to end on February 19, barring “unforeseen developments”, and confirmed all affected guests were being taken to hospitals.
David Abel, a friend of Mr Steele who is also a passenger on the ship, said Wendy is missing her husband “a great deal” but said they are in regular contact.
He told BBC Breakfast: “Alan’s been in touch with me a couple of times today. If all goes well, he’s being tested again in a couple of days’ time, and if he is clear they will test him again 48 hours later and if he still remains clear he will be released from the hospital.
“But where’s he going to go? He will not be allowed back on board the ship.”
Mr Abel said he thinks the process of quarantining is “essentially important”, but added: “I wish the UK government would send out a ship or a huge plane and bring us to the UK.
“I would far sooner be in the UK under a better controlled environment than being on board the ship, to tell you the truth.”
Meanwhile, some 150 Britons being flown back from the coronavirus-hit city of Wuhan on Sunday will be quarantined at a facility in Milton Keynes.
South Central Ambulance Service said Kents Hill Park, a conference centre and hotel, would be used to house the returning citizens after they landed at RAF Brize Norton.
The group will remain at the site in isolation for 14 days, it added.
Everyone boarding the plane in the Chinese city, which is the epicentre of the outbreak, will be assessed and will continue to be monitored after landing in the UK on Sunday morning.
On Tuesday, Foreign Secretary Dominic Raab said that the flight would be the final service chartered by the Foreign Office to bring UK nationals back from the Chinese city.
The ambulance service said the presence of the group in Milton Keynes, Buckinghamshire, does not present a risk to local people.
The first group of Britons who returned on a flight last month are continuing their period of quarantine at Arrowe Park Hospital on the Wirral.
“The local site has been chosen because it offers appropriate accommodation and other facilities for those coming back from Wuhan while they stay in Milton Keynes,” the ambulance service said.
The death toll in the coronavirus outbreak in mainland China rose to 722 on Saturday, while new cases jumped to 34,546.
The Department of Health and Social Care said that 620 people in the UK have been tested for coronavirus as of 2pm on Friday, with three cases confirmed.
It is understood that the third person in the UK to be diagnosed with coronavirus caught the illness in Singapore.
He is reported to be a middle-aged British man and is understood to be the first UK national to contract the disease.
The man is thought to have been diagnosed in Brighton and was transferred to St Thomas’ Hospital in London, where there is an infectious diseases unit, on Thursday afternoon.
Two other patients, who had recently travelled from China, are still being treated at the Royal Victoria Infirmary infectious diseases centre in Newcastle.
One is a student at the University of York, while the other is a family member.