Asked how he proposed to approve a motion for a second Brexit referendum, the Conservative Remainer argued Parliament would "assert its authority".
Eloise Todd, head of the anti-Brexit campaign group 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".
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.
"It has become increasingly clear to me that the withdrawal agreement, which is being finalised in Brussels and Whitehall even as I write, will be a awful mistake", he wrote.
She has lost all authority and is incapable of negotiating a Brexit deal within her own party, let alone with the EU.
Johnson - who had backed staying in the European Union in the 2016 referendum - is the younger brother of Boris Johnson, who spearheaded the pro-Brexit campaign and quit as foreign secretary in July over May's Brexit blueprint.
Calling Theresa May's Brexit deal "a awful mistake", the minister of state for transport quit his post, claiming the United Kingdom economy would suffer from May's Brexit agreement.
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Brexit negotiations have put Britain on the brink of the greatest crisis it has faced since the Second World War, Transport Minister Jo Johnson warned in his resignation letter.
Lidington has insisted the government remains confident it can get its Brexit deal through parliament, despite the Democratic Unionist party (DUP) warning it was prepared to vote it down.
Former Brexit secretary David Davis, who quit over Brexit in July, said Mr Johnson was right to describe the Government's proposals as a "travesty of Brexit".
His brother praised his decision, saying they were "united in dismay" at the PM's handling of the negotiations.
Boris Johnson tweeted supported for his brother's resignation.
In marked contrast to his brother, who remains firm that leaving the European Union without a deal would be no bad thing, and resigned because he believed the deal being negotiated was too soft, Mr. Jo Johnson called for a second referendum.
Giving his response on Twitter, Boris wrote: "Boundless admiration as ever for my brother Jo".
Labour Brexit spokeswoman Jenny Chapman asked: "How are we meant to trust this government to deliver a good deal for the country when we have a Brexit Secretary who doesn't even understand the very basics of Brexit?"