Brussels, Belgium (CNN) British Prime Minister Theresa May's efforts to save her faltering Brexit deal floundered Friday, as EU leaders sent her away empty-handed and a leading official described the state of the debate in the United Kingdom as "sometimes nebulous and imprecise."

After being forced to pull a vote on her deal in the House of Commons, May pleaded with EU leaders to add legal assurances that would assuage lawmakers furious over a crucial element, the so-called Irish backstop.

But after an apparently lackluster presentation by May, EU leaders rejected the demands -- all but killing any hope of a parliamentary breakthrough in London -- and instead stepped up plans for a no-deal Brexit.

Juncker calls May 'woman of great courage'

European Commission President Jean-Claude Juncker expressed admiration for May, and said that the EU had treated her with "much greater empathy and respect" than some of her own MPs.

"This is a woman of great courage. Doing her job in the best way possible. I'm supporting the way she's dealing with that matter," he said.

But while he said May had been "fighting hard and bravely," he expressed doubt that any deal would get through the British Parliament in its current form.

Juncker said British lawmakers have "deep distrust" when it comes to the EU, and warned that that's not a good basis for future relations.

"I was following second by second the debate in the House of Commons and I noted there is a deep mistrust in the House when it comes to the European Union," he said at the news conference Friday.

"The second after an approval by the two parliaments, the British one and the European one, we will start the negotiations on the future relations," Juncker said.

Juncker was also asked about a seemingly tense exchange with May that was captured on video.

"She thought I had criticized her when I said yesterday that the British position was nebulous," Juncker said.

He said he was not referring to May, but rather to the state of the debate in the United Kingdom, and that the misunderstanding was later cleared.

May to hold talks 'in the coming days'

At a news conference Friday, May acknowledged that members of Parliament "will require more assurances" than she has been able to wring from fellow EU leaders so far. She said she would be "holding talks in the coming days" in pursuit of those.

But she also insisted that both sides want to make it work, and said the conclusions issued by the other 27 EU leaders Thursday night were welcome and "take us forward."

May added that the EU had given its clearest statement yet that it had no intention of the backstop ever being necessary; that if it were necessary, it would be temporary; and that the EU was ready to work quickly with the United Kingdom to establish their future relationship.

The Prime Minister insisted she had been "crystal clear" about the assurances on the backstop required by British MPs, an apparent rebuff of Juncker's characterization of the debate as "sometimes nebulous."

Asked about an apparently frosty exchange with Juncker, May acknowledged they had had a "robust" discussion.

May's plea to EU leaders

Addressing EU leaders Thursday, May had urged them to provide guarantees on the Irish backstop, an insurance policy designed to avoid the return of customs posts on the Irish border. "We have to change the perception that the backstop could be a trap from which the UK could not escape," May told EU leaders, according to Downing Street.

She urged them to "trust me to do what is right," Downing Street said, saying it was in no one's interests to "run the risk of an accidental 'no-deal' with all the disruption that would bring."

May put forward a two-stage process -- a political declaration now, and a legally binding assurance in January, an EU diplomat told CNN.

But her plans did not go down well. Leaders were looking for two things from May Thursday: a concrete proposal to break the impasse and assurance that she could get it over the line in Parliament. They got neither, the diplomat said.

After May's presentation, said to have been short on specifics, the final text of the summit's conclusions was changed to cut a suggestion that the EU consider what further assurances could be provided to the UK, another EU diplomat said. A reference to contingency preparations for a no-deal exit was left intact.

May failed to persuade EU leaders to offer her a lifeline after a bruising no-confidence vote this week.

Some EU leaders urged British lawmakers directly to face up to their responsibilities. "Now MPs in London should be responsible and to know if they want to have the best possible deal, or to go in a direction where they don't know what will come out, " said Xavier Bettel, the Luxembourg Prime Minister.

"They should think about the interests of their voters, and of the people in their country, and if they respect that, they should vote for it," he said.

German Chancellor Angela Merkel and French President Emmanuel Macron also effectively put the ball in London's court. "As far as the EU is concerned we'll be doing everything in our power but we'll be expecting the United Kingdom to do everything in their power to make this happen," Merkel said in her closing remarks.

"There are a lot of concerns and fantasies around the backstop," Macron said. "What comes next is in the hands of the British Parliament."

Sebastian Kurz, the Austrian chancellor, who has been a key ally of the British Prime Minister, attempted to put a positive gloss on the outcome. "We told Theresa May once again that we will not open or reopen the withdrawal agreement, but besides the withdrawal agreement there is a huge understanding of both sides and a wish to find a way to deal with Brexit," he said.

In London, the opposition Labour Party called for May to reschedule the delayed Brexit vote in Parliament before Christmas.

"The last 24 hours have shown that Theresa May's Brexit deal is dead in the water," said Labour leader Jeremy Corbyn. "She's failed to deliver any meaningful changes. Rather than ploughing ahead and recklessly running down the clock, she needs to put her deal to a vote next week so Parliament can take back control."

Breaking the impasse?

Events of the past few months have solidified May's reputation as a political survivor. But the decision to call off the Brexit vote in Parliament, followed by a no-confidence vote in which a third of her MPs registered their disapproval of her leadership, have left May seriously weakened.

With just over 100 days before the March 29 Brexit deadline, there is precious little time to negotiate a new deal should the current one be voted down, even if the EU wanted to.

Labour and other opposition parties have pledged to call for a vote of no confidence in the government, should May's bill be defeated. That could lead to a general election.

Following the defeated Conservative leadership challenge this week, the math may be in their favor. May's critics within her own party are stuck with her through the Brexit process -- the failure to win this week's confidence vote means they can't launch another challenge for 12 months.

The only other option for May would be to call a second Brexit referendum. That would require her to reverse her implacable opposition to such a move -- although it would not be her first big policy U-turn.

After Wednesday's vote on her leadership, May told reporters she was ready to "get on with the job of delivering Brexit." Less than 48 hours later, she's looking more stuck than ever.

This story has been updated to correct the spelling of the Austrian chancellor's name and the number of days until the Brexit deadline.