The ex-Cabinet minister said: "We have opened ourselves to perpetual political blackmail".
"A suicide bomber murdered many in the courtyard of my office in Helmand", he wrote on Twitter.
With Monday marking just 200 days until the UK's exit from the EU, Mr Johnson's successor as Foreign Secretary Jeremy Hunt pleaded for Tories to get behind Mrs May and the Chequers plan.
MP Sarah Wollaston said she would consider leaving the party if Mr Johnson became leader.
"Some need to grow up".
I'm stunned at the nature of this attack.
Hardline Brexiteer Johnson, who quit as foreign secretary in July in protest at May's Chequers proposals for Britain's future relationship with the European Union (EU), was also rebuked by housing minister James Brokenshire for his latest attack.
Mr Tugendhat replied: "That's exactly my view and that is why I say I have made my view very clear".
Bernie Sanders Strikes at Amazon With New Tax Bill
For example, if an employee receives $1,000 in federal assistance, the employer will be charged that amount as a tax. The median Amazon worker , meanwhile, was paid $28,446 past year , according to company filings.
Backbench Tory MP Nadine Dorries said Johnson's critics were "terrified of his popular appeal".
In a reference to the policies of President Trump, Mr Johnson said: The US now boasts economic growth rates far in excess of this country, at about 4.5%, and with record low unemployment.
On the BBC's Andrew Marr Show, Mr Javid said: "I think there are much better ways to articulate your differences". I think that is to be admired and I think that puts public confidence in our political system that someone is speaking up for them.
Mrs May's spokesman said: "What we need at this time is serious leadership with a serious plan and that's exactly what the country has with this Prime Minister and this Brexit plan".
Mr Johnson used his regular Daily Telegraph column to set out an alternative economic plan, with the hope that tax cuts will provide a boost to the public finances. Grieve said Johnson's phrasing in the Mail on Sunday article was "entirely in character: crude but, for some, entertaining populist polemic."He said there was a lack of coherent thought in the article".
The announcement followed newspaper allegations the former London mayor had been having an extra-marital affair.
"I think that's reflected in the way we're being pushed around in the negotiations; we are the sixth biggest economy in the world, we are third or fourth biggest defence budget in the world, we are a world player, we've got the biggest soft power projection with the English language and our broadcast media in the world, and I think we've handled these negotiations particularly badly".
Some of the lurid details were said to have come from a dossier compiled by May's office in 2016 when she was competing with Johnson for the Conservative leadership.