She agreed with chairman of the Tory backbench 1922 Committee, Sir Graham Brady, that the process to choose a new leader should begin the week after she stands down.
Five other candidates have so far confirmed their intention to stand:
Foreign Secretary Jeremy Hunt
International Development Secretary Rory Stewart
Health Secretary Matt Hancock
Former Foreign Secretary Boris Johnson
Former Work and Pensions Secretary Esther McVey
Earlier, other Conservative leadership contenders clashed over Brexit.
Mr Hancock said Mrs May’s successor must be more “brutally honest” about the “trade-offs” required to get a deal through Parliament.
And Mr Stewart said he would not serve under rival Boris Johnson because of his backing for a no-deal exit.
In the Mail on Sunday article, Mr Raab – who voted for Brexit in the 2016 referendum – said: “We can’t live in a country where politicians make promises to respect your vote in a referendum, and then junk them if they don’t like the verdict.
“The country now feels stuck in the mud, humiliated by Brussels and incapable of finding a way forward. The prime minister has announced her resignation. It’s time for a new direction.”
Mr Raab said “leadership with conviction” was needed to change the dynamic of Brexit.
He added: “I will fight for a fairer deal on Brexit, a fairer deal for British workers, and a fairer society where every child can fulfil their potential.”
He was the second Brexit secretary to resign, in November 2018.
At the time he said he quit the cabinet over “fatal flaws” in the draft Brexit agreement with the EU.
In the Mail in Sunday interview, Mr Raab said in there was still time to negotiate changes to the Northern Ireland backstop – the agreement to maintain an open border on the island of Ireland in the event that the UK leaves the EU without securing an all-encompassing deal.
He described it as the “Backstop of EU laws”, over which he said the UK had no say over.
He said: “That is a reasonable, limited, request and would work in all sides’ interests.
“It is the only solution MPs have approved. But, we will not be taken seriously in Brussels, unless we are clear that we will walk away on World Trade Organisation (WTO) terms, if the EU doesn’t budge”
Mr Raab also set out plans to raise the amount workers can earn before they pay National Insurance, as well as a cut of a penny off the basic rate of income tax.
He said that apprenticeships, paternity leave and the environment would be priorities if he became Prime Minister.
Who are the Conservative members?
Most members of most parties in the UK are pretty middle-class. But Conservative Party members are the most middle-class of all: 86% fall into the ABC1 category.
Around a quarter of them are, or were, self-employed and nearly half of them work, or used to, in the private sector.
Nearly four out of 10 put their annual income at over £30,000, and one in 20 put it at over £100,000. As such, Tory members are considerably better-off than most voters.