UK Prime Minister Theresa May has said the EU must treat the UK with "respect" in Brexit negotiations.
In a statement at Downing Street she said for EU leaders to reject her plan with no alternative at this "late stage of negotiations" was "not acceptable".
A defiant Mrs May added: "I will not overturn the result of the referendum nor will I break up my country."
Mrs May's statement came after EU leaders said Chequers plan would not work, at a summit in Salzburg.
The prime minister said: "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.
"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. So we now need to hear from the EU what the real issues are and what their alternative is so we can discuss them. Until we do, we cannot make progress."
She said the two sides were still "a long way apart" on two big issues: the post-Brexit economic relationship between the UK and EU, and the "backstop" for the Irish border, if there is a delay in implementing that relationship.
The two options being offered by the EU for the long-term relationship - for the UK to stay in the European Economic Area and customs union or a basic free trade agreement - were not acceptable, she added.
The first would "make a mockery of the referendum" she said, while the second would mean Northern Ireland would be "permanently separated economically from the rest of the UK by a border down the Irish Sea."
Mrs May said no UK prime minister would ever agree to that: "If the EU believe I will, they are making a fundamental mistake."
She added either would be a "bad deal" and repeated that "no deal is better than a bad deal".
She said the best outcome would be to leave with a deal and the UK had put forward a third way - her Chequers plan, which Donald Tusk had rejected on the grounds it would undermine the single market.
"He didn't explain how in any detail or make any counter proposal, so we are at an impasse," she said.
On the issue of the backstop, she stressed it would be unacceptable to keep Northern Ireland in the customs union, which would mean "breaking up our country".
The prime minister moved to reassure EU citizens living in the UK that, in the event no deal can be reached, "your rights will be protected": "You are our friends, our neighbours, our colleagues. We want you to stay."
And she stressed the government would do "everything in our power to prevent a return to a hard border" between Northern Ireland and the Irish Republic.
She added that "no-one wants a good deal more than me" but she would not overturn the referendum result or break up the UK.
"We need serious engagement on resolving the two big problems in the negotiations. We stand ready."
The UK is due to leave the EU on 29 March 2019.
Mrs May says her plan for the UK and EU to share a "common rulebook" for goods, but not services, is the only credible way to avoid a hard border between Northern Ireland and the Republic of Ireland.
But it is opposed by some within her own party who argue it would compromise the UK's sovereignty. And it got a cool reception at this week's EU summit in Austria.
In a news conference, European Council President Donald Tusk said there were some "positive elements" in Mrs May's proposals, known as the Chequers plan.
But he said EU leaders had agreed that the proposals needed to be redrawn: "The suggested framework for economic co-operation will not work, not least because it is undermining the single market."
He followed it up by posting a photograph on Instagram of he and Mrs May looking at cakes with the caption: "A piece of cake, perhaps? Sorry, no cherries."
The EU has argued that the UK cannot "cherry-pick" elements from its rulebook.
Earlier Brexit Secretary Dominic Raab expressed doubt over how serious EU leaders were about the negotiations.
He told the BBC's Politics Live: "It did not feel like the reciprocation of the statesmanlike approach that she (Mrs May) has taken".
Brexit: Defiant words from PM may not be enough
Laura Kuenssberg
She didn't budge. And she was never going to.
After
her embarrassment yesterday at the hands of EU leaders, with restive backbenchers at her back, Theresa May was never going to use this highly unusual appearance in Downing Street and dramatically reverse out of her position.
Simply, to back down less than 24 hours later would have looked weak.
And the prime minister has invested huge amounts of time, effort, and political capital into her Chequers plan.
Ministers never believed that it would be the precise form of the final agreement.
But Number 10 believed that in good faith, they had put something credible on the table for discussion.
She has already lost two cabinet ministers over the compromise, and endured weeks of attacks from different wings of her party.
So one, albeit, highly embarrassing diplomatic bust-up was not going to force her to back down.
Her anger was visible though today, explaining again why she believes the two familiar post-Brexit trade options, Norway and Canada, cannot, and will not work for the UK.
And she called for respect from the EU: "Throughout this process I have treated the EU with nothing but respect, the UK expects the same".
A stern tone, strong words.
And in going further on citizens' rights too, perhaps Mrs May has sought to take the high ground, in contrast to what some in the UK are seeing as the EU's poor behaviour yesterday.
But while there is no remote sign from the PM today that she is about to compromise, forces in the EU and in her own party are intent on forcing her to do so.
Her problem is that they want to push her in different directions.
Rhetoric doesn't change the fact that few of the players involved outside Number 10 believe that the suggestions the prime minister has put forward can be the ones that ultimately will win the day.