In a video, Mr Javid said he wanted to “rebuild trust, to find unity and create new opportunities for our country”.
He said the results of the European elections – which saw his party score less than 10% of the total vote, compared to nearly 25% in 2014 – made it “all too clear” that the government “must get on and deliver Brexit to ensure there is renewed trust in our democracy”.
He added: “We must bridge divides to heal communities, reminding us of our shared values as a United Kingdom, and we must strengthen our society and economy so that everyone can benefit from the opportunities which a prosperous nation provides.”
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Mr Johnson wrote in the Daily Telegraph that voters had issued a “crushing rebuke” to the Conservatives, and the party could be “fired from running the country” if it does not deliver Brexit.
Mr Gove said “one message is clear” from the figures and that was the next Tory leader “absolutely needs to deliver Brexit”.
And Mrs Leadsom called the results “truly terrible”, saying they “demonstrate the damage that has been done to the Conservative Party”.
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, including Mr Johnson and Mr Raab.
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”.
Nominations for the leadership will close in the week commencing 10 June, and MPs will then whittle down the candidates to a final two, which the wider membership of the party can then vote for.