Matt Hancock, previously the Culture Secretary, has been named as the new Health Secretary.
Mr Johnson stepped down earlier on Monday, criticising Prime Minister Theresa May in his resignation letter, warning that Brexit is "dying" and Britain is "headed for the status of a colony". "I am deeply concerned at the position of the British government and urge them to embrace this vital issue".
Both Mr Johnson and Mr Davis had signed up on Friday to Mrs May's blueprint for Brexit at an all-day summit at Chequers on Friday.
The former management consultant was an ardent backer of Remain during the 2016 European Union referendum, warning that Brexit could "harm the NHS".
Writing on Twitter, he added: "Huge honour to be appointed Foreign Secretary at this critical moment in our country's history".
Under Conservative Party rules, a confidence vote in a leader can be triggered if 48 Conservative lawmakers write a letter requesting one.
"The world is looking at us as a country wondering what type of country we're going to be in a post-Brexit world".
Theresa May's government is in crisis.
Addressing parliament just minutes after her office announced that Johnson, the face of Brexit for many in Britain, had quit, May told MPs she appreciated the work of her two ministers.
He said: "My principal job is to stand 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".
"It seems to me we're giving too much away, too easily, and that's a unsafe strategy at this time", Davis said in a BBC radio interview Monday morning.
Serena Williams vs Camila Giorgi, Wimbledon 2018
Despite the two-year gap since their previous meeting, Williams professed to having watched 52nd-ranked Giorgi play "a lot". It is the first time in Wimbledon history not one of the top 10 women's seeds have reached the quarter-finals.
Croatia fear England's Sterling in W Cup semi
Gazza was of course one of the star players the last time England played in an World Cup semi-final back in 1990. It is only the second time that Croatia have reached the World Cup semi-finals.
Belgium star Hazard: I used to support France growing up
Raphael Varane and Benjamin Pavard have contributed for France , while Jan Vertonghen began Belgium's comeback win over Japan. This means that the club will have at least one representative in the 2018 FIFA World Cup final.
Meghan and Harry join the Queen at RAF centenary service
Harry and Meghan's engagements will include official meetings with Taoiseach Leo Varadkar and President Michael D Higgins. The Duke and Duchess of Cambridge and the Duke and Duchess of Sussex arrive at the Westminster Abbey service.