Standing outside her Downing Street official residence, May said she would form a government by relying on support from "friends" in Northern Ireland's Democratic Unionist Party and lead Britain in talks with the European Union to secure a successful deal.
Signed by 574,787people at the time of writing, the petition also calls for Theresa May to resign after she lost her parliamentary majority. "He's in a permanent leadership campaign so I am not sure it qualifies as news", he said.
The Conservatives won 318 seats - eight less than it needed in order to secure a majority.
The new Parliamentary arithmetic means the Conservatives will need other parties' backing to get any new laws passed - and Graham Brady, chairman of the influential 1922 Committee of backbench Tory MPs, predicted this would change their party's policy plans.
But the wooing of the DUP risks upsetting the political balance in Northern Ireland by aligning London more closely with the pro-British side in the divided province, where a power-sharing government with Irish nationalists is now suspended.
BREAK: DUP has NOT yet reached any agreement with the Tories.
The arrangement being sought is known as "confidence and supply", under which the Tories could rely on DUP support in key votes like Budgets and no-confidence motions.
Senior party figures have cautioned against any immediate leadership challenge, saying it would cause only further disruption as Britain prepares to start the Brexit talks as early as June 19.
Labour leader Jeremy Corbyn, riding a wave of acclaim for his party's unexpectedly strong showing, called on May to resign.
To stay in power, the Conservatives are seeking support from Northern Ireland's Democratic Unionist Party. All the most senior ministers - including Treasury chief Philip Hammond, Foreign Secretary Boris Johnson, Defense Secretary Michael Fallon and Home Secretary Amber Rudd - kept their jobs and there were few changes in the Cabinet lineup.
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The British PM's bet that she could strengthen her hand by crushing "a weak opposition Labour Party" backfired spectacularly on Thursday.
The election result saw Mrs May scramble to form a minority government to have any kind of Commons majority.
The protest was led by students and trade unionists who chanted slogans in favour of Labour leader Jeremy Corbyn, who has stunned the British political establishment with impressive election results.
"There has been a lot of hyperbole about the DUP since Thursday, a lot of things said, a lot of people who really don't know what we stand for", DUP leader Arlene Foster told Sky News on Sunday morning.
Senior Conservative Gavin Williamson is in Belfast for talks with the pro-Brexit party, a spokeswoman for Ms May's office said. "We want to do what is right for the whole of the UK and to bring stability to the government of the United Kingdom".
Speaking on the BBC's Question Time, he said: "She is playing fast and loose, on Brexit, on Margaret Thatcher's greatest achievement - the single market".
British Prime Minister Theresa May was fighting for survival on Saturday after a failed election gamble undermined her authority and plunged the country into a major political crisis days before talks to leave the European Union start.
Ruth Davidson, the Conservative leader in Scotland, said she had asked May for assurances that there would be no attack on gay rights after a deal with the DUP.
It was not immediately clear what the DUP's demands might be and one DUP lawmaker suggested support might come vote by vote.
Promises likely to be made by Mr Williamson and the prime minister are likely to concentrate on the Northern Ireland implications of Brexit.
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