British Prime Minister Theresa May has demanded that the European Union treat the United Kingdom with "respect" in the on-going Brexit negotiations, as she delivered a statement at Downing Street the day after the Salzburg summit, where EU leaders rejected her proposals for Britain's exit from the bloc, known as the Chequers plan.

May also defended the result of the 2016 public referendum which put Britain on the path to Brexit, and rejected any suggestion that Northern Ireland, part of the United Kingdom but not part of the British mainland, should remain in any kind of customs union post-Brexit.

"Throughout this process, I have treated the EU with nothing but respect. The UK expects the same. A good relationship at the end of this process depends on it," she said.

"At this late stage in the negotiations, it is not acceptable to simply reject the other side's proposals without a detailed explanation and counter proposals."

Her lack of success so far has seen some opponents call for a second referendum, or a public vote on whatever Brexit terms the government can come up with, but she defended the outcome of the 2016 vote, where 52 percent of those who took part voted for the country to leave the EU, an issue which has dominated British politics ever since.

"The referendum was the largest democratic exercise this country has ever undergone. To deny its legitimacy or frustrate its result threatens public trust in our democracy," she said.

One of the main sticking points of Brexit talks has been the issue of Northern Ireland, which shares a border with the Republic of Ireland, an EU member.

There is currently no hard border between the two territories, but this will be a major issue when Britain has left. However, May was quick to dismiss the EU's proposal to keep Northern Ireland inside any post-Brexit customs union. "As I have already said, that is unacceptable," she said. "We will never agree to it. It would mean breaking up our country. We will set out our alternative that preserves the integrity of the UK."

Among other British political parties, the Liberal Democrats were the first to respond to the statement. Tom Brake, the party's Brexit spokesman, said: "Instead of pontificating to television cameras, May must now recall Parliament to explain how she got the country into this terrible mess, what her plan is to get us out of it, and when we can have a people’s vote and an exit from Brexit."

May faces plenty of opposition to her Brexit proposals from within her own Conservative Party, most notably from a so-called hard Brexit group, known as the European Research Group.

"It is time for the government to start putting forward as its plan a Canada-style free trade agreement for the whole of the UK," said the group's chair, Jacob Rees-Mogg.

"This is the most realistic approach and similar to the EU's proposal. On the matter of the UK's border with Ireland, the EU can have the solutions proposed by the European Reform Group that respect the integrity of the single market and which have been supported by the Democratic Unionists (the Northern Irish party who support May's minority government)," he said.

Business leaders greeted May's announcement with caution. Carolyn Fairbairn, director of the Confederation of British Industry, said: "Rejection of Chequers helps nobody. Pragmatism must come before politics. Every day lost in rhetoric is lost investment and lost jobs."

Adam Marshall, director general of the British Chambers of Commerce, said: "While the Prime Minister's determination to reach a deal is appreciated, businesses tell us over and over that the time for posturing from both sides is over.

"Many firms are hugely worried about a messy and disorderly outcome, and the potential impact on their ability to trade and grow. Others could be caught flat-footed. Both sides must make every effort to avoid this scenario," he added.

Laura Kuenssberg, BBC's Political Editor, said May has demonstrated "stern tone, strong words", demanding respect, but she is not convinced that the rhetoric will do much to change the situation with the EU. Kuenssberg believes tougher tone will please some in her party, who criticized May for a lack of leadership.

On the currency markets, May's comment that negotiations with the EU had reached an "impasse" saw the pound suffer, being down 1.5 percent against the dollar at one point, and on course for its biggest one-day slump of the year so far.