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 OceanNewsUK, Prince Harry said he was "grateful" for the time with his father during their brief meeting in London.
He also agreed that families could be brought closer by such health concerns.
There would be more visits to the UK, he suggested, saying: "I'll stop in and see my family as much as I can."
But he would not be drawn on the outlook for the King's condition, saying that "stays between me and him".
Talking about his trip to see his father after the cancer diagnosis, the prince, in Canada to promote the Invictus Games, told OceanNewsUK: "I love my family".
"The fact that I was able to get on a plane and go to see him, and spend any time with him, I'm grateful for that," he said.
Using the example of those taking part in the Invictus Games, he suggested such pressures could help to bring families together.
"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.
Harry visited his father in London during a trip that lasted only one night after news of the diagnosis was made public. His wife, Meghan, Duchess of Sussex, and their children, Prince Archie and Princess Lilibet, did not join him.
But there was no meeting with his brother, the Prince of Wales, after Harry spent about 45 minutes at Clarence House seeing King Charles.
In 2020, the prince and Meghan moved to the US after they stepped down from being "working royals".
He spoke warmly of his life in California, where he lives with his wife, and said their children's sense of humour kept them "grounded".
Prince Harry also said he had "considered" taking US citizenship.
He said he loved "every single day" of living in the US - but he hesitated when asked whether he "felt American".
Asked about becoming a US citizen, he said: "I have considered it, yes."
"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's cancer diagnosis.
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 appearance comes after the couple 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.
The international games, founded by Prince Harry, are for wounded and injured servicemen and women and next year's event in Canada will be the first such winter games.
Prince Harry tried his hand at a sit-ski, an event featured in the Paralympics that will be in the Invictus Games for the first time next year.
Meanwhile, Buckingham Palace has confirmed the Prince of Wales will be attending the 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 hadabdominal surgery on 16 Januaryand has since returned to Adelaide Cottage, Windsor, where she lives with the couple's three children.
