Calling for a second referendum to be held on Brexit, Mr Johnson denounced the choice between her deal or no-deal as a "failure of British statecraft on a scale unseen since the Suez crisis" that had left Britain facing "vassalage" or "chaos".
"The prime minister has told colleagues this week we should aim to conclude the withdrawal agreement as soon as possible but we will not do that at any cost", PM May's office said in a statement.
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.
"This is a con on the British people: there is no evidence that the kind of Brexit that we've failed to negotiate while we are still members can be magically agreed once the United Kingdom has lost its seat at the table".
Tory MP Anna Soubry, a vociferous Remain campaigner, said she had "huge respect" for Mr Johnson, telling The Guardian: "Jo isn't the only minister who shares these views and I hope others will follow his lead".
Joseph Johnson believes that Britain is on the verge of severe crisis since the Second world war and assured the loyalty of the Conservative party, despite the resignation. Now his lesser-known brother Jo, who was also a government minister, has followed him out.
There are real questions about how we will be able to guarantee access to fresh food and medicine if the crucial Dover-Calais trade route is clogged up.
In a 2016 referendum on European Union membership, Britons voted 52-48 percent to leave the bloc.
His brother Boris, who quit as foreign secretary in July, praised his decision, saying the brothers were "united in dismay" at the Prime Minister's handling of the negotiations.
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". A second referendum on the departure is anathema to the supporters of Brexit.
"The gap is so wide now between what was promised at the time of the referendum and what is in the agreement currently proposed by the First minister that I did not had another choice than to submit my resignation to the First minister", he insists in this video.
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. Talks have stalled over the contentious Ireland border.
Its leader Arlene Foster said Friday she could not vote for the deal if it could see the province treated differently from the rest of the UK.