Hunt, the MP for South West Surrey, has served as Health Secretary since 2012.
Britain's Secretary of State for Foreign and Commonwealth Affairs Boris Johnson arrives at 10 Downing Street in London, Britain, July 3, 2018.
The Prime Minister said she was "sorry and a little surprised" over Johnson's resignation, after "productive discussions" on Friday.
The fallout from recent Brexit related resignations has triggered another round of musical chairs and resulted in the UK Government replacing Matt Hancock MP at the helm of the Department for Digital, Culture, Media & Sport (DCMS) with Jeremy Wright QC (MP for Kenilworth and Southam).
"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".
She said: "My concern is that the Chequers' proposal is an opening bid in negotiations and the European Union may well try to erode the position agreed by the cabinet".
His concluding thought on the issue: "If a country cannot pass a law to save the lives of female cyclists - when that proposal is supported at every level of UK Government - then I don't see how that country can truly be called independent".
Hunt, who had been the health secretary, is considered one of May's most loyal ministers.
Despite a "collective agreement" on the plan at Chequers last Friday the pair said they could not back the proposals, specifically the plan for a common rulebook for industrial goods and agricultural products.
"I like Boris Johnson".
Brexit minister Steve Baker and unpaid parliamentary aides Conor Burns and Chris Green also resigned. "That's certainly up to the people, not up to me". He said: "From what she has said Brexit does not mean Brexit and what we will end up with is the very worst of both worlds - still in the European Union but with no vote".
Dominic Raab, the former minister of state for housing and planning, was named the new Brexit secretary.
May's supporters worry that more ministers could resign, and pro-Brexit lawmakers from her Conservative Party could trigger a no-confidence vote in the prime minister.
The news has sparked speculation of a potential leadership challenge, and the threat of another general election.
Responding to her statement, Labour leader Jeremy Corbyn claimed there was a "crisis in Government" adding: "It is clear this Government is not capable of securing a deal to protect the economy, jobs and living standards".
The Council President told reporters: "Politicians come and go but the problems they've created for their people remain".
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