The British opposition Labour Party's decision to vote against the deal deepened their pessimism.
MPs have voted to delay Brexit beyond the scheduled date of March 29 in dramatic parliamentary scenes which saw the Conservative Party split down the middle.
One on Wednesday would decide whether Britain should simply leave on March 29 without any deal at all.
While a disorderly Brexit would hurt the EU, several top European officials warned that no delay would be granted unless Britain comes up with a clear and substantial plan, which it would try to achieve if given more time.
As president of the European Commission, the EU executive body, Juncker does not necessarily speak for the member state governments who have the final say on the terms of Britain's exit. This House will have to answer that question.
"I felt in all good conscience I had to vote tonight to clearly rule out any form of second referendum. Or does it want to leave with a deal, but not this deal?"
"These are unenviable choices", she said.
Labour ordered its MPs to abstain on a cross-party bid to delay Brexit to allow a referendum on backing whatever deal is agreed or remaining in the EU.
The US President said Brexit is "tearing countries apart" but it would be unfair to hold a second referendum.
The deal achieved in talks with Jean-Claude Juncker, the President of the European Commission (EC), was a "joint legally binding instrument" that will sit alongside the withdrawal agreement that would allow the United Kingdom to trigger a formal dispute against the EU if it tried to keep the United Kingdom within the backstop indefinitely.
Cox said he now considered that the legally-binding provisions of the new documents "reduce the risk that the United Kingdom could be indefinitely and involuntarily detained" within the backstop, "at least in so far as that situation had been brought about by the bad faith or want of best endeavours of the EU".
"The attorney general's advice is that the legal risk remains unchanged", Conservative MP Andrea Jenkyns tweeted.
After two-and-a-half years of haggling since the 2016 Brexit referendum, Juncker cautioned this was the last chance for Britain.
Labour's Brexit spokesperson Keir Starmer said Mr Cox confirmed that no "significant changes" had been secured and the government's strategy was "in tatters".
"Now is the time to come together to back this improved Brexit deal and to deliver on the instruction of the British people", she said.
"Even if May's deal is not passed today, the market consensus remains of the view that a hard Brexit will be avoided and this view continues to provide underlying support for the pound".
The British leader announced after the negotiations that she had secured the "legally binding changes" to the vexing issue of the Irish border that lawmakers had long-sought.
The vote puts the world's fifth largest economy in uncharted territory with no obvious way forward; exiting the European Union without a deal, delaying the March 29 divorce date, a snap election or even another referendum are all now possible.
It has also increased the chances that May's fragile government could fall, sparking a snap election.
"We recognise that the Prime Minister has made limited progress in her discussions with the European Union".
Will there be another general election?
More than two and a half years after the country voted to leave the European Union - and with no certainty about when or how it will - many Britons are simply fed up.