He described this as "a failure of British statecraft unseen since the Suez crisis" but said even a no-deal Brexit "may well be better than the never-ending purgatory" being put forward by the prime minister.
Johnson says Theresa May's Brexit strategy will make Britain subordinate to the EU.
Labour Shadow Brexit Minister Jenny Chapman said that Theresa May had "lost all authority and is incapable of negotiating a Brexit deal within her own party, let alone with the EU".
A government minister has quit with a stark warning over the impact on Kent of a no deal Brexit.
Downing Street said on Friday evening it would not agree to a second referendum vote under any circumstances and reiterated a promise not to sign the United Kingdom up to any deal which could return a hard border to Northern Ireland.
He said the country was "on the brink of the greatest crisis" since the second world war, and that Ms May's settlement left the United Kingdom with a choice of "vassalage or chaos".
"It has become increasingly clear to me that the withdrawal agreement, which is being finalised. even as I write, will be a awful mistake", he wrote in a stinging resignation statement.
He added voters had been offered two choices which represented "no choice at all".
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He added: "In the sense that if you betray the British people where they no longer believe in democracy... you don't know what the consequences are".
In an interview with Sky News on Saturday, Fox said he sees "a very hard end" to fraught discussions in which European Union negotiators think they're making too many concessions and British counterparts think they haven't won enough commitments to guarantee the nation's independence.
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 yesterday if he could agree with Johnson's call for a new referendum, Corbyn said: "Not really, no".
Johnson wants a three-way referendum giving the people a choice between remaining in the EU, May's deal and no deal.
"Boundless admiration as ever for my brother Jo", he said. "It does not remotely correspond to the mandate of the people in June 2016".
Like his older brother, Mr Johnson enjoyed a distinguished career in journalism before entering politics, working for the Financial Times as, among other roles, South East Asia bureau chief and Paris correspondent.
But calls for a second Brexit referendum have been attacked by ex-first secretary of state Damian Green.