The 44-year-old is a staunch Eurosceptic and was an outspoken campaigner for Leave ahead of the EU referendum.
Raab, the Esher & Walton MP, enters Cabinet for the first time, having previously served in more junior ministerial posts in the housing and justice departments.
He has previously clashed with Theresa May when he labelled feminists ‘obnoxious bigots’.
Raab waded into an equality row, claiming men were facing “flagrant discrimination”.
He said: ““While we have some of the toughest anti-discrimination laws in the world, we are blind to some of the most flagrant discrimination – against men.
“From the cradle to the grave, men are getting a raw deal. Men work longer hours, die earlier, but retire later than women. That won’t be fixed for another seven years.
“One reason women are left ‘holding the baby’ is anti-male discrimination in rights of maternity/paternity leave …
“Meanwhile, young boys are educationally disadvantaged compared to girls, and divorced or separated fathers are systematically ignored by the courts.”
He added: “Men caused the banking crisis. Men earn more because they are more assertive in pay negotiations,” he said. “One FT [Financial Times] commentator recently complained that: ‘High-flying women are programmed to go for high-flying men. Most men aren’t attracted to women who are more successful than they are.’
“Can you imagine the outrage if such trite generalisations were made about women, or other minorities? Feminists are now amongst the most obnoxious bigots.”
In 2011 during her time as Home Secretary May scolded Raab, saying that such offensive and sexists comments were “not the way forward” if he was striving to battle sexism in the workplace.
The Tory housing minister also hit the headlines when it was exposed that he was a member of a Facebook group that called for an end to council housing and suggested workhouses for the poor.
Raab said he was unaware he was part of the social media group, called British Ultra Liberal Youth. The group has previously called for the NHS to be privatised.
Following his resignation, Mr Davis said he did not want others to follow him out of Government but added it was a “decision of conscience” that must be made “in their own minds”.
He insisted a leadership challenge would be the “wrong thing to do” and ruled out standing against Mrs May.
Letters calling for a leadership contest have reportedly been submitted to the backbench 1922 Committee by some Conservatives over the weekend.
The Prime Minister faces a stormy meeting with Tory MPs and peers in Parliament on Monday evening as she tries to win support for her plan, which includes an offer to Brussels to share a “common rulebook” on goods and form a new UK-EU free trade area.
In his resignation letter, Mr Davis said the policy “hands control of large swathes of our economy to the EU and is certainly not returning control of our laws in any real sense”.
The responsibility for leading the negotiations should now go to an “enthusiastic believer in your approach, and not merely a reluctant conscript”, he said.
In her reply, Mrs May told him: “I do not agree with your characterisation of the policy we agreed on at Cabinet on Friday.”
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The PM is expected to use a Commons statement to tell MPs that the strategy agreed on at Chequers is the “right Brexit” for Britain.
Brexiteer Cabinet minister Michael Gove admitted the Chequers’ plan was not everything he had hoped for, but he was a “realist” and the Prime Minister’s lack of a Commons majority meant the “Parliamentary arithmetic” was a factor in deciding what could be adopted.
But all eyes are on Foreign Secretary Boris Johnson, who backed the proposals at Chequers, despite claiming that defending the plans was like “polishing a turd” during the meeting.
There had been confusion surrounding the position of Brexit minister Suella Braverman who had reportedly also quit, but a spokesman for the department confirmed she remains in post.
Labour leader Jeremy Corbyn said the Prime Minister “has no authority left and is incapable of delivering Brexit”.