Boris Johnson, the front-runner in the Tory leadership race, has said the party risks being “dismissed” if it does not deliver Brexit.
Writing in In the Daily Telegraph, he said voters in the EU election issued a “crushing rebuke” to the Conservatives.
Fellow candidate, Foreign Secretary Jeremy Hunt, said the party faces an “existential risk” over Brexit.
Eight candidates have declared they are standing for leader, after Theresa May said she would resign.
Mr Johnson said voters had issued the party with a “final warning” as the Tories came fourth in Hillingdon, where he is an MP, and the Brexit Party emerged with the largest number of MEPs overall.
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He said: “If we go on like this, we will be fired: dismissed from the job of running the country.”
Mr Hunt said on Twitter that the “painful result” meant there was an “existential risk to our party unless we now come together and get Brexit done”.
With some results still to declare, the overnight count has seen Conservative voters deserting the party in droves, with the party so far scoring less than 10% of the total vote – compared to nearly 25% in the last EU election.
BBC political editor Laura Kuenssberg said it was the worst performance for the Conservatives as a party since 1832.
Mr Johnson said the message from the results was clear for the Conservatives and the party risked a “permanent haemorrhage” of support.
The only way to avoid that outcome, he said, was to “come out of the EU; and that means doing it properly.”
The leadership race began when Mrs May announced she would stand down as Tory leader on 7 June, saying it was time for another prime minister to try to deliver Brexit.
So far eight candidates have said they want to run for the Tory leadership.
Environment secretary Michael Gove and former Brexit Secretary Dominic Raab declared on Sunday, joining Matt Hancock, Jeremy Hunt, Boris Johnson, Esther McVey, Andrea Leadsom and Rory Stewart.
Attitudes toward a no-deal Brexit are sharply divided, with several candidates saying they are prepared to let the UK leave the EU on the new deadline on 31 October without a deal if necessary.
In his Telegraph column, Mr Johnson said that “no-one sensible” would aim exclusively for a no-deal Brexit.
But he added that “no-one responsible would take no-deal off the table”.
Mr Raab told the BBC he would fight for a “fairer” Brexit deal with the EU – but if that were not possible, the UK would leave with no deal in October.
And Mr Gove confirmed he would run to “deliver Brexit” and unite the party.
Other candidates have stressed the need to get a Brexit deal passed in Parliament.
Writing in the Times, Health Secretary Matt Hancock said Conservatives had to deliver Brexit through Parliament, “whether we like it or not”.
He added: “The brutal truth is that plans that cannot command the confidence of parliament would risk a general election. We would be punished for our failure to deliver Brexit and under any leader this would risk Corbyn by Christmas.”