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.
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”.