"I don't think another vote would be possible because it would be very unfair to the people who won, who say, 'What do you mean you're going to take another vote?' That will be tough". "And it's a shame that it has to be that way, but I think we will stay right in our lane".
Trump's comments come just hours after he had seemingly handed the government a big boost, declaring in a tweet that a trade deal with the United Kingdom as having "unlimited" potential.
And he said that embattled British Prime Minister Theresa May, whose own party twice rejected her Brexit plan, would have already cut a deal if only she had listened to his advice.
"I think it could have been negotiated in a different manner, frankly", Trump said.
Trump said Thursday he was "surprised at how badly it has gone from the standpoint of negotiation", adding, "I gave the Prime Minister my ideas on how to negotiate it.
We can do a very big trade deal with the United Kingdom".
Britain is scheduled to leave the European Union on March 29.
Mr Trump told reporters he did not believe a second Brexit referendum could be justified, saying it would be "unfair" to the 52 percent of Britons who voted Leave.
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Trump invited his Irish counterpart, the prime minister, a position known in Ireland as taoiseach, to weigh in, and Varadkar noted that he had a "different opinion" from Trump. "I predicted it was going to happen".
Trump spoke hours before British lawmakers voted to delay Brexit for at least three months. He also said his administration will negotiate a new "America first" deal with the European Union or the bloc will face tariffs.
Mr Trump has said he will be visiting the Republic of Ireland at some point this year.
Varadkar, sitting alongside Trump at the White House, said he looked forward to discussing Brexit with Trump and that he would like to see a European trade deal with the United States.
During a press conference with the taoiseach, the U.S. president confirmed he would make the trip because Ireland is "a special place". "But they're going now and that's their decision", Varadkar said, adding that the negotiations shouldn't cause any problems in Northern Ireland.
"We talked about Brexit, something that is turning out to be a little more complex than they thought it would be".
"What I've asked for is an understanding of our situation, particularly when it comes to Northern Ireland and avoiding a hard Border and protecting the peace process".