He attacked the PM for presiding over "the largest defeat in the history of our democracy" on her Brexit deal before saying there has been no offer of all-party talks from the prime minister to break the impasse. She invited lawmakers to start meeting with her to find a solution.
Gardiner, a former aide to Margaret Thatcher, said a second referendum would be a "slap on the face" to the estimated 17.4 million British people who voted to leave the EU.
Jeremy Corbyn, leader of the opposition Labour Party, has proposed a motion of no-confidence in May's government, which will be debated and put to a vote on Wednesday.
The Democratic Unionist Party (DUP) - the Northern Irish party that props up May's minority government - said it would work with the Prime Minister to secure a better deal.
May's Brexit agreement was created to keep trade rules between the world's fifth-biggest economy and its largest export market nearly unchanged for a transition period running through the end of 2020.
While May withstood the challenge to her leadership, it is yet another indication that Brexit has thrown British politics into chaos.
In the lead-up to Tuesday's crushing defeat, May's deal was criticized by politicians and pundits on all sides of the political spectrum.
Sterling was changing hands at $1.2848, slightly below $1.2860 seen late Tuesday in NY.
He added that 2011's Fixed-term Parliaments Act "was never meant to prop up a zombie government", saying that the prime minister had "lost control" and suffered an "historic and humiliating defeat".
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Downing Street said May will come back to parliament with a new Brexit proposal on Monday.
The vote against May's Brexit deal was largely expected but the severity of the defeat took some by surprise: the House of Commons voted 432-202 against the deal, with 118 of May's fellow Conservatives turning against her. On Tuesday evening, British lawmakers rejected the deal Prime Minister Theresa May had reached with the European Union, opening a Pandora's box of possible outcomes to be considered in the weeks to come.
Also at stake is the future of May as the prime minister, when MPs begin to vote on the 599-page withdrawal agreement that she reached with Brussels after almost three years of tough talks.
Meanwhile, the former leader and co-founder of UKIP, Nigel Farage, said Wednesday it was freakish May had not resigned as prime minister following her defeat.
The no-confidence vote was supported by all of Britain's opposition parties, the BBC notes, including the Scottish National Party and the Liberal Democrats. Britain is due to leave the European Union on March 29. But given the European Union's negotiating stance, there is little she can do to make the deal more palatable to its opponents.
As well as Mr Tusk's tweet, there has been plenty of comment on Tuesday's vote from across Europe.
Corbyn said that May's Conservative Party was "fundamentally split" on Brexit meaning it was unable to govern the country at a time of crisis.
"A hard Brexit will be the worst of all options", she said.
But her measured remarks contrasted with the blunt message from French President Emmanuel Macron, who told Britons to "figure it out yourselves". Now, they said, May could stop pretending that her deal had a chance of passing and move to discussing more realistic options. Brookings' Amanda Sloat has been tracking the Brexit process closely; in previous posts, she has explained what May's plan entails, the revolt within her party, and why the plan didn't get a vote in December. Business groups have expressed wide alarm at the prospect of a no-deal exit.
The EU and European governments have warned that the British parliament's rejection of a Brexit deal heightens the risk of a disorderly withdrawal from the bloc. "A new plan is needed immediately".