The Duke of Sussex has told US TV that he "jumped on a plane" as soon as possible to see King Charles after his father was diagnosed with cancer.
In an interview with ABC News, Prince Harry said he was "grateful" for the time with his father during their brief meeting in London.
But he wouldn't be drawn on the outlook for the King's condition, saying that "stays between me and him".
Prince Harry also said he had "considered" taking US citizenship.
He said he loved "every single day" of living in the United States and that he had considered applying for citizenship, but at the moment it wasn't a priority.
The prince told ABC's Good Morning America that "I love my family" and agreed that families could be brought closer by such health concerns.
"Throughout all these families I see it on a day-to-day basis - the strength of the family unit coming together," he said. "Any illness, any sickness, brings families together," he said.
There would be more visits to the UK, he suggested, saying: "I'll stop in and see my family as much as I can."
The prince spoke warmly of his life in California, where he lives with his wife Meghan, and said that their children's sense of humour kept them "grounded".
Prince Harry hesitated over being asked whether he "felt American" and was asked whether he would consider becoming a US citizen.
"I have considered it, yes" he replied, prompting the reporter to ask what had stopped him.
"I have no idea... I am here... American citizenship is a thought that has crossed my mind but certainly it is not a high priority right now," he said.
Prince Harry was speaking in his first interview since the King postponed all public-facing duties because of his cancer diagnosis.
In 2020, the prince moved to the US with his wife, Meghan Markle, Duchess of Sussex, after they stepped back as senior royals.
Prince Harry visited London on 6 February without Meghan and their children, Prince Archie and Princess Lilibet, less than 24 hours after the announcement about the King's health.
But there was no meeting with his brother, the Prince of Wales, after Harry spent about 45 minutes at Clarence House seeing King Charles.
King Charles has paused all public-facing duties while he receives treatment, with senior royals - including Queen Camilla and the Prince of Wales - taking on his duties for some events.
The Good Morning America interview comes after the couple also relaunched their Archewell website, the name of their foundation.
It has been rebranded the Office of Prince Harry and Meghan, the Duke and Duchess of Sussex.
On Wednesday, the couple visited the training camp for the 2025 Invictus Games in Whistler, British Columbia.
Prince Harry tried his hand at a sit-ski, an event featured in the Paralympics which will be seen in next year's Invictus Games for the first time.
Meanwhile, Buckingham Palace has confirmed the Prince of Wales will be attending the British Academy of Film and Television Arts (Bafta) film awards in London on Sunday.
For Prince William, president of Bafta, this will be a second day of royal engagements since the King's unspecified cancer diagnosis and his wife's operation.
The Princess of Wales had abdominal surgery on 16 January and has since returned to Adelaide Cottage, Windsor, where she lives with the couple's three children.