Mr Hayes, who originally indicated his support for former London mayor Boris Johnson to be the next Prime Minister, said: "The country voted for Britain to exit from the European Union and the only person who can take that forward is the person who believed in it".
In a detailed speech setting out his leadership vision, Mr Gove said he stood by all of the promises made by Vote Leave during the referendum contest.
Gove was one of the prominent campaigners for the U.K.to leave the European Union in the run-up to last week's referendum, which saw voters opt for Britain to exit, or Brexit, the bloc.
His intervention comes amid widespread delight at the top of the Scottish Conservatives at Boris Johnson pulling out of the race after being ambushed by his former Leave colleague, Michael Gove.
As well as the support of Mr Fallon, Mrs May has also received the backing of Transport Secretary Patrick McLoughlin and the Daily Mail newspaper. They will then go to a vote of the wider party membership.
Within couple of hours after Gove entered the fray, Johnson announced his withdrawal saying he had too little support among MPs to stand for the leadership.
North East Tory MP Guy Opperman has praised all the front-runners in the battle to become Conservative leader - and says he hasn't yet chosen who to back.
"I came in the last few days reluctantly and firmly to the conclusion that while Boris has great attributes, he was not capable of uniting that team and leading the party and the country in the way that I would have hoped".
"I came to realise this week that, for all Boris' formidable talents, he was not the right person for the task".
"I think she's the only one who is really going to deliver what we want", Paterson said.
Mr Gove drew attention to the poor perception of him from certain quarters of the public.
During the campaign, Gove dismissed all such warnings as "Project Fear" and said that "people in this country have had enough of experts".
"I enjoyed working with Boris during the referendum campaign, I think he has great talents and great abilities but you need something else to be prime minister, you need that grit, that executive authority, that sense of objective, that clarity".
Casting himself as a reluctant candidate, Gove said "I did nearly everything I could not to be a candidate for leadership of this party".
Gove is up against May and three other candidates to succeed Cameron as party leader and prime minister.
She told ITV's Good Morning Britain: "In the end I felt it was better to put my own name forward, because you do need a choice of candidates and it seemed to me that we might end up with only one candidate who had actually supported the Leave campaign. And her track record shows that when Theresa arrives in Brussels, Europe's bosses sit up and listen".