His departure followed that of Brexit Secretary David Davis and several junior figures.
Culture Secretary Matt Hancock replaces Mr Hunt as health secretary.
The UK is due to leave the European Union on 29 March 2019, but the two sides have yet to agree how trade will work between the UK and the EU afterwards.
There have been differences within the Conservative Party over what shape Brexit should take – although on Friday, at the prime minister’s country retreat at Chequers, the cabinet reached a “collective” agreement on proposals for the future relationship between the EU and UK.
In his first comments as foreign secretary, Mr Hunt said he would be standing “four square” behind the prime minister “so that we can get through an agreement with the European Union based on what was agreed by the Cabinet last week at Chequers.
“This is a time when the world is looking at us as a country, wondering what type of country we are going to be in a post-Brexit world.
“What I want to say to them is Britain is going to be a dependable ally, a country that stands up for the values that matter to the people of this country, and will be a strong confident voice in the world.”
Mr Johnson said in his resignation letter that Mrs May’s proposals for post-Brexit trade would leave Britain a “colony” of the EU.
The Brexit “dream is dying, suffocated by needless self-doubt”, he said.
Instead, “we appear to be heading for a semi-Brexit, with large parts of the economy still locked in the EU system, but with no UK control over that system.”
Mrs May said she was “sorry – and a little surprised” by Mr Johnson’s move after his apparent support on Friday.
She said the deal agreed by the cabinet after their “productive discussions” would “honour the result of the referendum” and allow the UK to “take back control of our borders, our law and our money”.
The prime minister earlier faced down backbench critics at a meeting of the 1922 committee, amid rumours they were close to getting the 48 signatures needed to trigger a no-confidence vote that could have removed her as prime minister.
Jeremy Hunt, who has been health secretary for the past six years and recently secured a £20bn boost in funding for the NHS, was a Remain campaigner in the 2016 EU referendum. He has since said he is a convert to the Brexit cause.
His appointment follows the promotion of staunch Leave supporter Dominic Raab to the cabinet as Brexit secretary – but the BBC’s political editor Laura Kuenssberg said most of the top jobs in government were now held by ministers who had backed Remain.
There are potentially dozens of Tory rebels who want to halt Mrs May’s Chequers compromise but it is not clear how committed they are, added our correspondent, and Monday’s resignations might have left her with a more pliable and supportive cabinet.
Her future now depends on how Brussels reacts to her Brexit proposals – and what Mr Davis and Mr Johnson plan to do next, she added.
Both men had signed up on Friday to Mrs May’s blueprint for Brexit – but 48 hours later, first Mr Davis and then Mr Johnson said they could not commit themselves to promoting the plans as they did not believe in them.
Brexit minister Steve Baker and two ministerial aides also resigned.
After his appointment as Health and Social Care Secretary was announced by Downing Street, Mr Hancock said: “Really looking forward to joining @DHSCgovuk at such an important time for our great NHS. I can’t wait to get started.”
Attorney General Jeremy Wright is the new culture secretary, with backbencher Geoffrey Cox replacing him as attorney general.
Who is Matt Hancock?
West Suffolk MP since 2010 and former Bank of England economist
Was a close ally and former chief of staff to then chancellor George Osborne
Supported Remain in the EU referendum
The 39-year-old father-of-three launched his own smartphone app after becoming culture secretary to better “connect” with his constituents
What about Jeremy Wright?
Former criminal lawyer who was first elected to Parliament in 2005 and currently represents Kenilworth and Southam
Attorney General for England and Wales Advocate General for Northern Ireland since July 2014
The 45-year-old father-of-two lists his interests as travel, especially to the United States, music, James Bond films and playing golf badly