EU leaders have praised Theresa May’s “courage” and “determination”, but French President Emmanuel Macron also urged “swift clarification” on Brexit.
Mr Macron said Mrs May had done “courageous work” on Brexit, in his response to her resignation statement.
But he stressed the need to “maintain the smooth functioning of the EU, which requires swift clarification”.
Earlier, EU Commission President Jean-Claude Juncker said the EU had “failed” over the UK’s 2016 vote to exit the EU.
“Here we failed, because we didn’t adopt the position that was necessary. Abstention is not a position,” he told the German public broadcaster ARD.
Prime Minister May’s withdrawal agreement – reached with the EU in November after arduous negotiations – has been rejected three times by Westminster MPs.
‘Woman of courage’
Mr Juncker said there was “no change” in the EU position on that agreement, commenting on Mrs May’s decision on Friday to step down on 7 June as Conservative Party leader.
He described her as “a woman of courage” for whom he “has great respect”, his spokesperson said.
EU chief Brexit negotiator Michel Barnier also voiced “full respect” for Mrs May, “for her determination, as Prime Minister, in working towards the UK’s orderly withdrawal from the EU”.
The withdrawal agreement’s terms for keeping the Northern Ireland border open and free of physical infrastructure – the so-called “backstop” – are a major stumbling block for the UK parliament.
Irish Taoiseach (prime minister) Leo Varadkar has won solid support from fellow EU leaders on the backstop.
He thanked Mrs May for “agreeing with us to retain and strengthen the Common Travel Area”, which ensures reciprocal rights for Irish and British citizens in each other’s countries, “as though we were citizens of both”.
That arrangement, he said, “will withstand Brexit whatever form it takes”.
EU respect tinged with frustration
By Adam Fleming, BBC News, Brussels
The EU establishment, like everyone, marvelled at Theresa May’s amazing ability to stay standing.
“If you’re a lion tamer you’re going to get bitten,” said one diplomat this morning. They were grateful that she respected the rules of the negotiations and didn’t rock the boat on other EU business.
EU leaders would bolster her position when things got tough – a photo-op here, some complimentary words there. But they weren’t prepared to compromise on the big one – the backstop to prevent a hard border between Ireland and Northern Ireland – because they think it’s already a compromise.
They think it took her too long to realise – or admit – that the UK’s economy required something that looked like a customs union. And they were amazed at her regular misreading of her own party and parliament.
EU Brexit negotiators have been war-gaming potential replacements for weeks and the scenario that seems to have been considered most seriously is a Prime Minister Boris Johnson asking for changes to the backstop that cannot be granted, and then blaming the EU.
But there’s been no formal discussion about what comes next. That will probably happen at a summit of leaders next week, supposedly about appointments to the EU’s top jobs and now inevitably about Brexit.
Juncker’s Brexit regrets
Speaking before Mrs May announced her resignation, Mr Juncker asked: “How could anybody else achieve what she couldn’t?”
“If you tell people for 40 or 45 years ‘we’re in it, but not really in it’, we’re part-time Europeans and we don’t like these full-time Europeans, then you should not be surprised if people follow simple slogans once they’re asked to vote in a referendum.”
The UK’s previous prime minister, David Cameron, clashed with Mr Juncker over the EU budget and other issues before arguing – unsuccessfully – to stay in the EU in the 2016 referendum.
EU institutional shake-up
Mr Juncker will be replaced by a new Commission chief – not yet chosen – in November. New leaders will be chosen for all the EU institutions after the 23-26 May European elections.
He belongs to the centre-right European People’s Party, the bloc which won the last European elections in 2014.
He voiced hope that the UK would leave the EU by 31 October – the new deadline set by EU leaders. The UK did not meet the planned 29 March deadline as exit terms had not been ratified.
Mr Juncker denied that the UK Brexit vote was a personal defeat for him.
“Nobody listens to me in Britain anyway. They should, but they don’t. There was nobody in Britain who confronted the lie with the incontrovertible truth,” he said.