MPs will hold another set of non-binding votes on various Brexit options in the Commons on Monday.
The Draft Withdrawal Agreement put forward by the prime minister fails to "take back control" across a plethora of key areas and does not deliver the Brexit Havering overwhelmingly voted for.
She said the implications of the vote were "grave", adding: "I fear we are reaching the limits of this process in this House".
And if it doesn't pass at this fourth attempt, well then we are into mega crisis territory because at that juncture the PM would be mandated by MPs to negotiate with the EU a customs-union-based Brexit.
This option commits the government to negotiating "a permanent and comprehensive UK-wide customs union with the EU" as part of any Brexit deal.
Alistair Burt, who resigned as foreign office minister after voting against May's Brexit deal, told Sky News he didn't think a new election would be helpful.
"The Remainers clearly want a customs union as an alternative but there isn't a majority for that in the party - it would destroy the [Tory] party", one ministerial source told The Telegraph.
After 170 Tory MPs - including 10 members of the Cabinet - wrote to Mrs May urging her to take the United Kingdom out of the European Union quickly as possible, Mr Gauke reiterated he would resign rather than support a no-deal break.
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May had already offered to resign before the next election anyway, and her latest offer came with one rather large condition: that lawmakers ratify her Brexit deal - the same deal that, in January, those very same lawmakers defeated by the biggest margin in the U.K.'s recorded parliamentary history.
This gives the public a vote to approve any Brexit deal passed by Parliament, before it can be implemented. Opposition parties are baying for an early general election.
Sir John said the best way forward is "not to wreck the British economy", the country's worldwide reputation or constitutional relationships via a no-deal Brexit, adding the customs union and single market additions are the potential compromise.
Thousands gathered outside Parliament to protest against the delay, bringing traffic to a standstill.
Ex-Prime Minister Sir John Major warned Mrs May against calling an election to end the impasse and said the country may need a time-limited "national government" including both Tory and Labour politicians.
To avoid that, Mrs May can try to seek a further Brexit extension from the European Union, which is likely to require voting for MEPS in new European Union elections.
"We would end up with a Ramsay MacDonald-esque government where we would be completely at odds with our own party", referring to the first-ever Labour prime minister - who ultimately ended up having to govern with on behalf of the Tories, after losing the support of his own backbenchers. But with the Labour Party and the Democratic Unionist Party opposed to the plan, a majority vote was unachievable.
"We are clearly going to have to consider very carefully the will of parliament", Gauke said.