S&D group MEP Richard Corbett has predicted that there will be growing calls in the UK for a "rethink" on Brexit.
He told a debate in Parliament on Wednesday that many British would demand a review of the decision "once they realise the reality of what it entails and how these contrast with the promises they were given."
Corbett said the UK decision to quit the EU had thrown up a "multitude" of issues and complexities. It had, he said, left the UK government with several "unpalatable" choices and had "huge implications" for the EU.
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He was speaking in a debate on a state of the union address by Commission President Jean-Claude Juncker in Strasbourg.
"This," said Corbett, "is going to be the mother of all divorce cases but, on both sides, we have to caution against being too hasty."
Corbett said the UK government had not yet triggered article 50 "because it has no idea what to do" in light of the decision to leave the EU.
"We have got to make the best of a terrible situation but I think that once they realise the reality as opposed to the promises of what Brexit entails, they will want to think again."
Scottish nationalist MEP Alyn Smith told the debate, "Once people realise that the promises of the Leave campaign cannot be met the whole thing will unravel."
However, following Juncker's speech, Ukip MEP Nigel Farage said, "I am pleased the British voted to leave the EU."
Farage, who sits close to Juncker in the chamber, told him, "It is clear that no lessons have yet been learned from Brexit because you are still talking about 'more Europe.'"
He was also critical of the UK government, saying it "ought to get on" with the task of formally exiting the EU.
Front National MEP Marine Le Pen condemned Juncker's "insipid" speech.
She said, "Rarely have I heard such an insipid and faulty speech. You would have thought it was about a funeral but for the EU.
"What Brexit did was to break a taboo and, contrary to what we were warned, the UK is doing alright after the vote. The apocalypse in the UK that was forecast has not happened but, still people like Juncker show disdain for the result. It is time to move on."
ECR group leader Syed Kamall urged the EU to "take heed" of the Brexit vote, saying, "otherwise, the voices of discontent will be heard across Europe and do not be surprised if other similar referendums follow."
He was also critical of Juncker's performance, telling MEPs, "It was billed as a new start but all we got was the same old mantra about more military integration and requests for more money from member states. It is always more, more, more."