"What Theresa May was trying to do with her proposal last week was to come up with a compromise that would keep all sides on ball within the Conservative party, and clearly that has failed", Clark said.
The prime minister has warned the Tory party it must unite or face the prospect of Jeremy Corbyn in power.
The ex-Foreign Secretary left the foreign secretary's official residence at Carlton Gardens, central London, late in the evening, accompanied by his wife Marina Wheeler, and without speaking to waiting reporters.
May said: "What we are proposing is challenging for the EU".
Johnson followed Brexit Secretary David Davis out the door as a hard-won government consensus on future trade ties with the bloc disintegrated less than three days after it was forged, and nine months before Britain is due to leave the EU. A third member of the government, Steve Baker, a junior minister in Davis' Department for Exiting the European Union, also resigned.
He went on to explain how the current political turmoil in Britain will work to the EU's advantage in Brexit negotiations.More news: Brazil arrive back in Rio de Janeiro following quarter-final defeat
More news: Awaiting Trump’s Supreme Court Nominee
More news: At least one killed in Virginia helicopter crash
Mrs May is expected to tell MPs that the strategy agreed on at Chequers is the "right Brexit" for Britain.
"The trouble is that I have practised the words over the weekend and I find they stick in the throat".
Labour leader Jeremy Corbyn said Mrs May should make way for a Labour administration if her Government can not get its act together quickly.
Under Conservative Party rules, a confidence vote in a leader can be triggered if 48 Conservative lawmakers write a letter requesting one.
Matt Hancock is the new health secretary.
In response to questions about whether the government is in "meltdown", May's spokesman said "it is not".
In her statement to Parliament, May attempted to give an impression of business as usual, saying she would present a detailed paper onThursday on the government's negotiating position for a post-Brexit relationship with the EU.