Donald Trump’s ambassador to Britain has insisted that the US president “admires and respects” Theresa May, following the transatlantic spat over Trump sharing anti-Muslim propaganda online.

Woody Johnson used an interview with Bloomberg on Friday to attempt to calm the diplomatic storm created by Trump’s retweets of videos from the far-right group Britain First, which prompted Britain’s ambassador to Washington complain to the White House.

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“The president and the prime minister have a very, very good relationship. I know the president admires and respects the prime minister greatly,” Johnson said.

“I know that the prime minister was one of the first, the first visitor to the White House after the inauguration.”

He added: “My job and the president’s job is to protect Americans. He’s doing the best that he can. You’re going to have little stumbles along the road. Absolutely. You’re going to have things that happen. But the intent is there and it’s genuine, and it’s going to happen.”

May criticised Trump’s sharing of the extremist material, saying it had been “wrong”. But the president followed up his initial retweets with a riposte to the prime minister in which told her to concentrate on fighting terrorism in the UK.

Politicians from across Britain’s main parties lined up to criticise Trump’s behaviour on Thursday, with many calling for the offer of a state visit to be formally withdrawn.

May passed on an invitation from the Queen for a full state visit when she visited Trump at the White House days after his inauguration. The invitation was accepted – but privately few in government expect it to take place in the foreseeable future.

The home secretary, Amber Rudd, condemned Trump’s retweets on Thursday, but urged MPs to focus on the “bigger picture” of Britain’s relationship with the US over the long term.

“As home secretary I can tell the house that the importance of the relationship between our countries, the unparalleled sharing of intelligence between our countries, it has undoubtedly saved British lives. That is the bigger picture here and I would urge people to remember that,” she said.

Johnson took a similar stance in his conciliatory interview on Friday, saying: “Our special relationship with the UK is centuries old and it’s been well tested through thick and thin, but it’s based on trust.

“And trust is something that you earn over a long period of time. And I think trust is 100% there and so we will get past all these things and get on with our main job and leadership in the world, which needs leadership.”

Johnson, a billionaire Republican donor and owner of the New York Jets NFL team, also claimed he liked the weather in the UK, finding it to be better than he had expected.