In an article sent to journalists, the MP for Orpington said he would vote against the withdrawal agreement which the prime minister was trying to agree with the European Union, describing it as "a bad mistake".
Asked whether other ministers should quit over the issue, Mr Johnson told BBC Radio 4's Today programme: "I think this is so important that it's up to MPs to take a stand".
Downing Street said last night 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.
Former Foreign Secretary Boris Johnson's brother and Orpington MP underlined that the country will be offered an agreement that will "leave our country economically weakened, with no say in the European Union rules it must follow and years of uncertainty for business", or no deal, "that I know as a transport minister will inflict untold damage on our nation".
The plan emerged after Jo Johnson, a transport minister, resigned from the Government over Brexit and vowed to vote against Mrs May's deal in the House of Commons.
The MP for Orpington, who left his post as Transport Minister on Friday, said that those in government will be "reflecting hard" about whether or not to follow in his footsteps.
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Tory former deputy prime minister Lord Heseltine, who backs the People's Vote campaign for another referendum, said: "This is a very significant resignation".
"This is the time for all of them, as Jo Johnson has done, to look deep inside their consciences, to reflect deeply on their responsibilities, and do the right thing for future generations".
But he said a second referendum is "not the way forward and is not supported by the public".
Calling for a new referendum, Mr Johnson said: "My view is that this is so different from what was billed that it would be an absolute travesty if we do not go back to the people and ask them if they actually do want to exit the European Union on this extraordinarily hopeless basis".
She said: "Huge respect for Jo Johnson".