Sarah Elliott, chairwoman of Republicans Overseas UK, told CNN she did not think Trump would be fazed by the stunt.
He claims that Brexit voters may not get "what they voted for".
Mr Smyth, who is originally from Ireland, said Trump fans had been given a bad rap, and there had been lots of support for the pub's temporary renaming.
"I would say Brexit is Brexit", Trump said at North Atlantic Treaty Organisation press conference.
"It's an easy target, those people who are visibly different", they said.
Asked about Johnson's hardline negotiating techniques, Trump said the former minister "was right".
Supporters of Brexit say such a trade deal with the world's biggest economy would be one of the great benefits of exiting the bloc.
"It's also an opportunity to tear down the bureaucratic barriers that frustrate business leaders on both sides of the Atlantic".
May is trying to unify her deeply divided Conservative Party behind her Brexit plans with some of her own lawmakers openly speaking of a leadership challenge.
"There is no stronger alliance than that of our special relationship with the United States and there will be no alliance more important in the years ahead", the British leader said.
May said there is "no stronger alliance" than the special relationship between the two countries and "there will be no alliance more important in the years ahead".
May reached out to him just days after Mr. Trump was inaugurated, extending the invitation for a coveted state visit that would be hosted by Queen Elizabeth II, but that gesture has proved far more controversial than expected as Mr. Trump has shown little interest in maintaining "diplomacy as usual" with European allies.
Gorgeous said that they have worked closely with the Metropolitan Police and other march organisers - along with the people behind London's planned LGBTQ+ Community Centre, which are providing stewards - to co-ordinate the day.
"There is definitely a sense that Britain needs America more because of Brexit", said Bronwen Maddox, head of the Institute for Government, a think tank.
The Police Federation has warned the visit will put "unquestionable pressure" on United Kingdom police forces as tens of thousands of people are expected to protest against the president in London on Thursday and Friday - and in Glasgow on Friday.
A high metal fence has been erected around the USA ambassador's central London residence, where Trump will spend Thursday night.
President Trump is expected to avoid areas such as central London where the majority of demonstrators could gather.
A campaign is also underway to get "American Idiot", a 2004 song by the USA punk rock band Green Day, to the top of the UK Singles Chart.
US President Donald Trump and his wife, Melania, have been being greeted with pomp and circumstance, along with protesters, as they arrived at Blenheim Palace for a gala dinner with British PM Theresa May.