"To present the nation with a choice between two deeply unattractive outcomes, vassalage and chaos, is a failure of British statecraft on a scale unseen since the Suez crisis", he wrote, adding that from the work of his own Department he was aware that a no-deal scenario could threaten the country's access to fresh food and medicine.
The Conservative MP added that it would be a "democratic travesty" if the public did not get to have their say once again. The EU insists that a deal must lead to either a soft Brexit for the entire United Kingdom, or a sea border between Great Britain and Northern Ireland.
Johnson wants a three-way referendum giving the people a choice between remaining in the EU, May's deal and no deal. "We will not under any circumstances have a second referendum", Downing Street said.
Transport minister Jo Johnson resigns over Theresa May's handling of Brexit.
His resignation was published after May spent much of the day in France and Belgium, laying wreaths alongside fellow leaders to mark the centenary of the end of World War One.
As negotiations with Brussels enter their final fraught stage, May's approach is under fire from all sides of the divisive Brexit debate.
A government minister has quit with a stark warning over the impact on Kent of a no deal Brexit.
But Mrs May will then have to convince many in her own party, as well as others, she has negotiated a good deal before Britain's planned European Union exit next March.
Boris Johnson, who himself resigned over May's Brexit plans, backed his brother's decision, saying: "We may not have agreed about Brexit but we are united in dismay at the intellectually and politically indefensible of the UK position".
"The second option is a "no deal" Brexit that I know as a transport minister will inflict untold damage on our nation".
His resignation comes as it is thought that a special cabinet meeting could be held early next week for ministers to approve the government's draft agreement on the terms of the UK's exit.
But May has drawn the fury of her Northern Irish allies the Democratic Unionist Party.
Eloise Todd, head of the anti-Brexit campaign group Best for Britain, said: "This is an incredibly fearless move from Jo Johnson at a time when the public desperately needs more MPs to act in the national interest".
Asked by German magazine Der Spiegel if he would stop Brexit if he could, Corbyn said: "We can't stop it".