Johnson confirmed he now plans to vote against any withdrawal agreement May brings before parliament.
With great regret, I'm resigning from the Government - I have set out my reasons in this article and the video below.
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.
Britain's junior Brexit minister told a German media outlet he was optimistic that Britain and the European Union would reach a Brexit deal soon, adding that he hoped this would happen in the next three weeks.
Boris Johnson, a leading supporter of Brexit who quit as foreign secretary in July, praised his brother's decision, saying they were "united in dismay" - despite their opposing views on Brexit - over May's handling of the negotiations.
He said the public should be given the chance to vote again on whether they want Brexit, and if so whether they accept what is on offer or choose to leave without a deal.
On Friday the DUP, whose support Theresa May relies on for votes in the Commons, said they can not support any deal which included the possibility that Northern Ireland would be treated differently from the rest of the UK.
The MP, who is Boris Johnson's brother, said the withdrawal deal now being negotiated with the European Union "will be a awful mistake".
Johnson's statement was published after May spent much of the day in France and Belgium, laying wreaths alongside fellow leaders to mark the centenary of the Armistice.
Eloise Todd, of Best for Britain, said: "This is an incredibly courageous move from Jo Johnson at a time when the public desperately needs more MPs to act in the national interest".
Calling for a second referendum, the Orpington MP said: "Given that the reality of Brexit has turned out to be so far from what was once promised, the democratic thing to do is to give the public the final say".
The transport minister Jo Johnson has resigned from the government over Brexit.
Cabinet ministers have been visiting 10 Downing St this week to see details of the government's plan for the "divorce deal" and subsequent trading and customs relationship with the EU.
Boris Johnson tweeted supported for his brother's resignation.
His older brother Boris wrote on Twitter after the announcement: 'Boundless admiration as ever for my brother Jo.
Britain's government cannot stop a case seeking to determine whether the country can unilaterally reverse Brexit - due to be considered by Europe's top court on November 27 - Scotland's Court of Session ruled on Thursday.