"We will be, instead of being in Europe but not run by Europe, we will be out of Europe and yet wholly subject to European rules". Mr Johnson believes the current deal would lead to an "incoherent" Brexit.
The departing minister noted Brexit had divided Britain, political parties and "families too", but added: "what is now being proposed won't be anything like what was promised two years ago" during the referendum campaign. "Now is the time for people to stand up for what they believe in or we will sleepwalk to a #Brexit disaster".
Theresa May has said the withdrawal deal is 95% done - but there is no agreement yet on how to guarantee no hard border in Northern Ireland.
"It has become increasingly clear to me that the withdrawal agreement, which is being finalised. will be a bad mistake", Jo Johnson wrote in a stinging resignation statement.
"The second option is a "no deal" Brexit that I know as a transport minister will inflict untold damage on our nation".
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.
In the June 2016 referendum, 17.4 million voters, or 51.9%, backed leaving the European Union while 16.1 million, or 48.1%, backed staying.
In a tweet, Boris said he had "boundless admiration" for his brother and that they are "united in dismay" at the indefensibility of May's position even though they disagree over Brexit.
Support for May's Brexit plan is dwindling, with the DUP today said to be furious over so-called "backstop to the backstop" proposals on the Irish border, which could divide the province from the rest of the United Kingdom on customs.
Jo Johnson's resignation, and his call for a fresh referendum to test voter sentiment, further complicates matters for May as she tries to strike a deal with European Union leaders that would pass muster with her own cabinet and win backing in Parliament.
The ex-transport minister said: "Hopes for the easiest deal in history have proved to be delusions".
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The anti-Brexit petitioners aim to show that Britain has a legal unilateral option of staying in the world's biggest trading bloc once it is known what the outcome of Brexit negotiations is.
Barnier is now attempting to negotiate a deal with Prime Minister Theresa May's government for Britain's "orderly" exit, which will take place in March next year.
"Theresa May is in office, but not in power".
They also suggested he had been privately supporting the work of anti-Brexit campaign group, People's Vote, "for some time" despite being a government minister.