The Tories were left tantalizingly short of an overall majority after the leader's gamble on a snap election backfired.
She added: "The mandate given to us by the people will be used responsibly".
"Theresa May is a dead woman walking".
Former Finance Minister George Osborne, a rival who was sidelined by May a year ago when she took over from David Cameron, told the BBC: "Theresa May is a dead woman walking". Brexit, he said, would amount to a "new partnership with Europe".
The Conservatives now plan to reach a so-called confidence and supply agreement with the DUP, which would involve it supporting a Conservative minority government on key votes in parliament but not forming a formal coalition.
The DUP is a socially conservative party that opposes abortion and gay marriage, and many of its members also are sceptical about man-made climate change and reject the theory of evolution.
A deal between the government and the DUP could also unsettle the precarious balance between Northern Ireland's British loyalist and Irish nationalist parties, whose power-sharing administration in Belfast collapsed earlier this year.
Asked whether a Conservative-DUP deal would endanger the peace agreement, Irish Foreign Minister Charlie Flanagan told ITV: "Not necessarily the case".
Arlene Foster said she hoped to capitalise on opportunities the situation presented for Northern Ireland.
"One piece of good news is the whole election has put pay to a hard Brexit", Osborne told CNN.
But the prime minister said she had a busy schedule ahead, with a Cabinet meeting on Monday and talks with French President Emmanuel Macron on Tuesday.
A's and Rays split rare scheduled doubleheader
The last time the Rays' had five straight quality starts was July 19 though 30, 2016 when the ripped off 10 in a row. C Wilson Ramos (recovery from torn ACL) may soon have his rehab upgraded from Class A Charlotte to Triple-A Durham.
The Queen's Speech - scheduled for June 19 - is read by the Queen at the opening of the Parliament and lists major policies and programmes that the elected government plans to undertake in the coming year.
But Anna Soubry, a Conservative member of parliament who campaigned ahead of last year's referendum for Britain to stay in the European Union, disagreed.
Theresa May's bid to do a deal with the Democratic Unionist Party to save her premiership has been plunged into uncertainty.
But he has responded in typically colourful fashion, tweeting: "Mail on Sunday tripe - I am backing Theresa may".
Downing Street initially said on Saturday that an outline agreement on a "confidence and supply" arrangement had been reached with the DUP which would be put to the Cabinet for discussion on Monday.
"This is not the time for sharks to be circling".
But Soubry said May's time in the top job would be limited. I think she will have to go, unfortunately.
Foreign Secretary Mr Johnson used an article in The Sun to stress his support for the Prime Minister: "To those that say the PM should step down, or that we need another election or even - God help us - a second referendum, I say come off it. Get a grip, everyone". "As a party, we are committed to securing the future growth of farming and food in Northern Ireland".
Defence Secretary Sir Michael Fallon said senior ministers had made clear to Mrs May she operated in the light of the election result.
May appeared isolated after her two closest aides resigned, paying the price for a disastrous electoral performance that cost the prime minister her majority. "This is still on", Corbyn told the Sunday Mirror newspaper.
On the other hand, Labour that did much better than expected under the leadership of Jeremy Corbyn claim to be the real winners and were ready to lead a minority government, had May stepped down.