"Our UK friends need to say what they want instead of asking us what we want," European Commission president Jean-Claude Juncker told reporters after the meeting. "Theresa May has led a courageous fight but unfortunately we are not seeing the results."
May survived a no-confidence vote in her own parliamentary party before the summit but has been greatly weakened by the revolt and the fact that she had to put off a vote on the deal because she knew it would not be passed.
She went to Brussels to plead for EU leaders' help to make it more palatable to Brexiters in her own Conservative Party and Northern Ireland's Democratic Unionist Party, whose support she needs to remain in power.
They claim that the temporary customs union designed to prevent the creation of a hard border between the Irish Republic and Northern Ireland, which is part of the UK.
May told EU leaders they must help her "change the perception that the backstop could be a trap from which the UK cannot escape", adding, "Until we do, the deal -- our deal -- is at risk."
But, she insisted, "With the right assurances, this deal can be passed. Indeed it is the only deal that is capable of getting through my parliament."
The atmosphere at the meeting was "tense", a source told the AFP news agency, with German Chancellor Angela Merkel and other leaders repeatedly interrupting May to ask exactly what she wanted.
A statement issued after the meeting confirmed the backstop would only "apply temporarily" and that EU members would work to "conclude expeditiously" the new trade deal.
But it is not legally binding and a phrase saying that the EU "stands ready to examine whether any further assurances can be provided" was removed after appearing in an earlier draft.
Diplomats said they did not want to imply that the deal could be renegotiated.
Juncker did promise that talks on the future trading relationship -- a deal on which would remove the need for the backstop -- would start as soon as MPs and the European Parliament approved the Brexit deal, however.
The summit continued Friday with the intention of agreeing a eurozone reform that is less ambitious than French President Emmanuel Macron wished.