However, senior DUP figures made clear they were looking at a limited "confidence and supply" arrangement - rather than a more formal coalition - leading to some MPs to predict that there could another general election before the year is out.
"They are wrong and they need to understand why we take those positions from a faith point of view and why we want to protect the definition of marriage", she said.
The DUP backs the continuation of the "triple lock" on retirement payouts, which the Tories had proposed to ditch.
Earlier, Mrs May announced that she meant to carry on in No 10 at the head of a minority government with the support of the Democratic Unionists (DUP) after falling eight seats short of an overall majority.
"That's what people voted for last June".
The Conservative Party failed to get an overall majority in yesterday's General Election, and must now try to secure an agreement with another party in order to govern.
"I asked for a categoric assurance that if any deal or scoping deal was done with the DUP there would be absolutely no rescission of LGBTI rights in the rest of the United Kingdom, in Great Britain, and that we would use any influence that we had to advance LGBTI rights in Northern Ireland".
SNP's loss has been Labour and Conservative Party's gain.
The shock result thrust the DUP into the role of kingmaker, with its 10 seats giving May a fragile but workable partnership.
The Prime Minister claimed that the Conservatives and DUP have together "enjoyed a strong relationship over many years".
A spokesperson for the British Pregnancy Advisory service, one of the largest abortion providers in the United Kingdom, said it was "troubling" that May would choose to work with the DUP despite its "regressive" views on women.
Corbyn would be unlikely to win backing for a minority government, but was clearly reveling in a storming performance after pundits had pronounced the Labour Party all but dead.
"We'll be working with the government, and all elected parties and MPs, to ensure that together we can make the changes we've called for in our manifesto a reality".
Leaders of other political parties were already calling for May to quit, and pundits were already speculating on who might succeed her should that happen.
'I will now form a Government - a Government that can provide certainty and lead Britain forward at this critical time for our country, ' she said.
"What tonight is about is the rejection of Theresa May's version of extreme Brexit", said Keir Starmer, Labour's policy chief on Brexit, saying his party wanted to retain the benefits of the European single market and customs union.
DUP leader Arlene Foster wants to avoid a hard border with Ireland and has spoken against a "hard Brexit".