His remarks came as the Prime Minister’s political difficulties at home showed no signs of abating on Sunday despite her Brexit plans winning the approval of the 27 EU leaders at a special summit in Brussels.
The DUP leader Arlene Foster, whose 10 MPs prop up Mrs May’s administration, said there were no circumstances under which her party would vote for the deal in its current form when it comes before the House of Commons, while Tory heavyweight and former leader Iain Duncan Smith said it would be “very, very difficult” to back the deal.
The 27 EU leaders unanimously backed Mrs May’s Brexit withdrawal deal and a political declaration on the future relationship between the European Union and the UK at the summit in Brussels on Sunday morning.
She now faces the fight of her political life as she spends the next fortnight campaigning for the deal across the UK before a vote in Parliament likely to take place in the second week of December.
Speaking after the agreement, European Commission president Jean-Claude Juncker said "it is a very sad day" and appeared to warn MPs in the UK that there would be no alternative if they voted down the deal in Parliament in December.
He said: "This is the best deal possible for Britain, this is the best deal possible for Europe.
"This is the only deal possible."
Mr Juncker added: "I am totally convinced that this is the only deal possible. Those who think by rejecting the deal that they would have a better deal will be disappointed in the first seconds after the rejection of this deal."
The European Commission president said "divorce is a tragic moment".
"Payments have to be made but the future understanding is one which has to be constructed. I don't think Britain will be a third country like other third countries are third countries. There is, between us, something which has the remainings of love."
He went on: "A country leaving the EU doesn't give rise to the raising of Champagne glasses or applause. It is a sad day and everybody who spoke today during the European Council attempted to express their sadness.
"It was broadly shared, if not unanimously shared."
He acknowledged that "ultimately we did have a problem, a difficulty, with our Spanish friends" at the end of the process over Gibraltar.
"I would point out today that the agreement we have obtained here today is an agreement which is good for Spain, so we are with Spain," he said.
Meanwhile, back in the UK Ms Foster was calling for a Brexit “third way”, insisting that the choice is not between the current draft agreement and no deal.
As EU leaders endorsed the proposed withdrawal treaty in Brussels, Mrs Foster made clear there were no circumstances under which her party would vote for it when it comes before the House of Commons.
"I believe we should use the time now to look for a third way, a different way, a better way," she said.
She added: "We shouldn't just accept an outcome for the sake of it - we should try to get a deal that is good for everybody."
European Council president Donald Tusk said: "Ahead of us is the difficult process of ratification as well as further negotiations.
"But regardless of how it will all end, one thing is certain: we will remain friends until the end of days. And one day longer."
Appearing The Andrew Marr Show on BBC One, Mrs Foster said the current deal's Irish 'backstop' - a measure that would align Northern Ireland with a number of EU regulations in the event a wider trade deal failed to materialise - was not acceptable.
"We're not advocating a no deal, we're advocating a better deal," she said.
"We want to see the Irish backstop gone. There is no need for the Irish backstop so let's get rid of it."
Mrs Foster insisted the DUP's confidence and supply agreement with the Conservatives remained "live".
But she again warned that her party would "review" the arrangement if the Brexit deal gained parliamentary support, though stopped short of saying she would definitely end the relationship with the Tories in those circumstances.
Former Tory leader Mr Duncan Smith told Sky News’ Sophie Ridge that it would be "very, very difficult" to support the Prime Minister's deal arguing that "far too much has been given to the EU."
"Well I don't want to stay in the EU. I campaigned and voted to leave the EU. I don't believe that, so far, this deal delivers on what the British people really voted for, take back control of your borders, your laws, your money. I think it has ceded too much control,” he said.
"I believe the Government needs to go back and say things like 'the backstop, we simply cannot agree it and you must take all of that reference out' and things to do with the Court of Justice. That would make this a better deal, but right now the balance is definitely tilted against this being a deal, I'm afraid, that delivers on what the Government said they would deliver, which is leaving the EU and setting out to be able to do our own trade deals.
"We won't be able to do our own trade deals if we're bound into the customs union, that is single one of the most important things that people voted for."
Mrs May released a near-800 word letter on Saturday night pleading with the country to get behind her deal. She said that Brexit day on March 29, 2019 needed to mark the moment in which the country left the labels of "Leave" and "Remain" behind and united as one people.
But former Labour leader Tony Blair said that another referendum was the "only way you are going to unite the country."
He told BBC's Andrew Marr Show people should choose between the sort of "proper Brexit" advocated by Boris Johnson or remaining in the EU.
Mr Blair, a prominent pro-EU campaigner, said he expected Labour to eventually back another vote.
Asked if there was a majority in Parliament for his position he said: "Not yet but I think it will get there."
He added: "I think it's moving that way and I would be really surprised if the Labour Party doesn't end up in the position of supporting another vote.
"Because there's no other proposition that can get through Parliament."
And Labour MP Lisa Nandy told Sky News that the Political Declaration "offers no guide as to what the future holds for the UK", adding: "There's been no attempt to reach out to Labour MPs like me... It's just too big a gamble for MPs like me to take with our constituents' futures."
Asked if she would be supporting the deal, she said: "Well I'd hoped that it would be, but in all honesty no it's not and it's inconceivable now that when this comes before Parliament in just a few days time that I'll be voting for it. I won't be voting to support the Withdrawal Agreement."
She added: "There is a real prospect now of no deal because many of those Leave voters, contrary to the opinion in Westminster, actually dislike this deal more than the Remain voters who contact me on a regular basis."