"I can still be prime minister".
On Brexit, the DUP supports leaving the European Union but opposes a return to a "hard" border with Ireland - which could happen if May carries through her threat to walk away from the talks rather than accept a "bad deal".
The 40-year-old says he remains "100 per cent committed" to Theresa May.
The Downing St. resignations came as May worked to fill jobs in her minority government and replace ministers who lost their seats on Thursday. "We will not be providing a running commentary".
Theresa May's advisers Nick Timothy and Fiona Hill are pictured outside Conservative Party Headquaters on June 9, 2017 in London, England.
He said he regretted introducing the controversial changes to social care without including "a ceiling as well as a floor" but denied the dementia tax was a "personal pet project".
The Prime Minister was forced to perform an unprecedented U-turn within days of the publication of the Tory manifesto by announcing that there would be a cap on social care costs, something that had been absent in the original policy document. Timothy said in a blog published on the ConservativeHome website that he had stood down on Friday and he took responsibility for the policy program in the Conservative manifesto.
Timothy, who was appointed as the chief of staff in July 2016, also clarified that "the weird media reports about my own role in the policy's inclusion are wrong: it had been the subject of many months of work within Whitehall, and it was not my personal pet project". Writing in the Times Perrior added: "For two people who have never achieved elected office, I was staggered at the disrespect they showed on a daily basis".
"Our two parties have enjoyed a strong relationship over many years, and this gives me the confidence to believe that we will be able to work together in the interests of the whole United Kingdom". Beleaguered May is appointing new members of her government after several.
David Davis will also stay on as Brexit secretary and Sir Michael Fallon will keep his role as defence secretary.
The deal sits uneasily with some Conservatives because of the DUP's opposition to abortion and same-sex marriage. German Chancellor Angela Merkel said she saw no reason to grant the United Kingdom a grace period.
James Cleverly, Tory MP for Braintree, told the BBC it was "probably the right decision" and "the honorable thing to do".
She needs to conclude a deal with the DUP in the next week or so ahead of the Queen's Speech, which will set out the new government's agenda. "Those principles won't be compromised".
Her hasty reappointment of some of the big guns in Cabinet should buy her some time, but Mr Johnson and Ms Rudd were strangely absent from the media in the hours after the election result became clear.
Conservative lawmaker Anna Soubry said May should "consider her position", while another, Heidi Allen, said she may not last six months.
Katie Perrior, who quit as May's communications chief in April, said Timothy and Hill were "great street fighters but poor political leaders" and exercised too much power over the prime minister.
May's party fell short of an overall majority following Thursday's vote, and plans to work with Northern Ireland's Democratic Unionist Party.
"I feel really bad for her today because, yes, she takes responsibility for this as she will because she has a real sense of public service and duty that runs right through her".