The Clacton MP, who defected from the Tories in 2014, resigned from the Ukip on Saturday, claiming he no longer needed to be associated with it as Brexit was “certain”.
During an interview with LBC on Monday, the newly independent Mr Carswell lauded the praise of Mrs May, but stopped short of suggesting he was willing to return to the Tories at the next general election.
Speaking to Iain Dale, he said: “The fundamental reason I’m leaving Ukip is because – job done."
Douglas Carswell lauded the praises of Theresa May during a radio interview on Monday
She is the real deal on this, she will deliver
He continued: “Ukip was established in 1993. I remember I was an undergraduate at university and I heard about the creation of this party, I was tempted by it then, 23 years later job done.
“Of course there’s going to be a debate about [Brexit], but the direction of travel is only going to end one way.
“On this, and I speak as someone who is a May-sceptic, she is the real deal on this, she will deliver.”
Mr Carswell has come under fire for his stubbornness not to call a by-election following his resignation from the party.
Senior Ukip figures, including former leader Nigel Farage, have demanded a new ballot take place in the Essex seaside town.
Douglas Carswell defected to Ukip in 2014 after serving as a Conservative MP
However, the independent MP said there was “no need” to go to the polls again as he was not “switching parties”.
“Quite often in politics, we cast aspersions on people that they may change their mind,” Mr Carswell continued.
“But, I think it’s extraordinary that people doubt my motives when I’ve been absolutely consistent on this.”
Brexit: Results of how the UK voted Mon, March 20, 2017
Much of the North East of the United Kingdom voted to leave the European Union including Sunderland, Gateshead, Darlington, Durham, Hartlepool, Middlesbrough, Stockton, Redcar and Cleveland, North Tyneside and South Tyneside, and Northumberland
Getty Images 1 of 8
GREAT YARMOUTH: The town of Great Yarmouth on the East Coast of England voted by 72% to leave the European Union.
Following the news of his departure from Ukip, some local voters branded Mr Carswell a “traitor”.
Speaking to the BBC, one angry man said: “He should be out, there should be a new by-election because he keeps changing from one thing to another.
“He doesn’t do what he’s asked to do, you try and find him own here and you can’t find him, so why should he be an MP?”
Responding to the criticism, Mr Carswell told LBC: “I always have enormous respect for people whether they’re critical or not.
“If they live in Clacton I take their views seriously, if they don’t, I thank them for expressing their views.”