Speaking on BBC Breakfast, he called the Royal Family one of the "great assets" of the UK.
He said: “My view on this is very straightforward: I am a massive fan – like most of our viewers – of the Queen and the royal family as a fantastic asset for our country.
“I’m absolutely confident that they are going to sort this out.”
Mr Johnson also refused to be drawn on whether Meghan had been subjected to racist media coverage.
Asked if the royals had the right to a private life, Mr Johnson said: “I think that all those dilemmas are well understood and I have absolutely no doubt they are going to sort it out.
“But they are going to sort it out much more easily without a running commentary from politicians.”
In response to suggestions that race was an issue in the media coverage of the Duchess of Sussex, Mr Johnson dodged the question, saying: “I don’t think this is helped by running commentary from politicians.
“The royal family is a fantastic part of our country – and so is the media, by the way.”
His comments come a day after a Sandringham summit where the Queen gave her backing to Harry and Meghan's new "independent life" away from full-time royal duties and they will begin a transition period living in the UK and Canada.
The Queen also stressed the discussions were “complex matters” for her family to resolve but she wanted “final decisions” in the “coming days”.