He added the alternative of no-deal would "inflict untold damage on our nation".
To give the public a choice between these two disastrous versions of Brexit would be a "failure of British statecraft on a scale unseen since the Suez crisis", he said, referencing the 1956 conflict against Egyptian President Gamal Abdel Nasser that is widely seen as marking the moment at which Britain formally lost its imperial role in the world.
It casts further doubt on Mrs May's chances of pushing her Brexit deal through the Commons.
"It can not be what you wanted nor did the 2016 referendum provide any mandate for it".
"On this most crucial of questions, I believe it is entirely right to go back to the people and ask them to confirm their decision to leave the European Union and, if they choose to do that, to give them the final say on whether we leave with the prime minister's deal or without it", he said, in his article.
"This is a fascinating situation in which Jo and his sister are united in opposing their brother Boris and his Brexit plans".
He wrote: "Boundless admiration as ever for my brother Jo".
His brother Boris Johnson whose promises in the Brexit campaign Jo may well have been referring to, praised Jo's resignation, saying they were "united in dismay" at the PM's handling of the negotiations.
In an open letter today he wrote: " The government may have to take control of prioritising which lorries and which goods are allowed in and out of the country, an extraordinary and surely unworkable intervention for a government in an advanced capitalist economy. It is a surrender of control.
Eloise Todd, of 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".
"We are barrelling towards an incoherent Brexit that is going to leave us trapped in a subordinate relationship to the European Union, with no say over the rules that will govern huge swathes of our economy".
"Well done to Jo Johnson for joining the ranks of those who think we can do better".
The resignation of such a high-profile figure, just days after Britain suggested it was close to a deal is seen as hugely significant. potentially leading to other Conservative MPs and Ministers speaking out in favour of a second referendum.
A Downing Street spokesman insisted: "The referendum in 2016 was the biggest democratic exercise in this country's history".
But he said a second referendum is "not the way forward and is not supported by the public".
His older brother Boris resigned in July to protest May's Brexit plan - but Boris did so as a staunch support of a hard-line Brexit, while Jo Johnson backed the "remain" side during the June, 2016 referendum.