DUP leader Arlene Foster has invited US President Donald Trump to visit Northern Ireland for The Open Championship.
On the prospect of another referendum, Mr Trump said: "I don't think another vote would be possible because it would be very unfair to the people that won".
"I lived in a country where if I'd tried to be myself at the time, it would have ended up breaking laws", he said, The Advocate reports.
When Mr Trump was asked about it by United States reporters, he turned to Mr Varadkar and said: "We have a border situation also".
"I'm not going to comment on Brexit". More than 30 million Americans, or about one in 10, have Irish ancestry.
This evening he will attend the annual Irish Fellowship St. Patrick's Day dinner, attended by over 1,000 Irish Americans.
He reminded the crowd that the shamrock tradition began nearly 70 years ago when Ireland's first ambassador to the United States, John Hearne, gave then USA president Harry Truman a small box of it.
Mr Trump spoke during the lunch
In his remarks at the vice president's residence on Thursday, Varadkar spoke about the change that has happened in his country, from homosexuality being illegal to a gay man being elected prime minister.
Trump was game to keep the ritual going despite his own outsider status and disdain for stuffy political convention. He also said Pence and his wife told him that he and his partner were "very welcome to visit their home in the future".
Thursday's events carried on despite awkward timing. We want to make sure there is trade between Britain and Ireland. He also said his administration will negotiate a new "America first" deal with the European Union or the bloc will face tariffs.
"The potential is unlimited!" he tweeted.
"Great commitment amongst the USA administration to help Northern Ireland".
The Irish PM said the most pressing issue facing his country was how to settle questions about the future of the border between Ireland, an European Union member, and Northern Ireland, which won't be.
"I look forward to talking to you later about Brexit, giving you our perspective on it and the real importance of protecting the Good Friday Agreement and the really hard-won peace in Northern Ireland", Varadkar told Trump, referring to the 1998 British-Irish peace accord.