The Clacton politician will now stand as an independent after he resigned from the eurosceptic party on Saturday.
Despite pressures from senior Ukip figures, including former leader Nigel Farage, Mr Carswell has insisted he will not call a by-election because he “will not be switching parties”.
In 2014, Mr Carswell defected from the Conservatives to Ukip, triggering a by-election which he subsequently won.
However, voters in the Essex seaside resort are now demanding another vote to be held because Mr Carswell is “no longer the person we voted for”.
Clacton voters were left fuming after Douglas Carswell quit Ukip on Saturday
I think he’s a traitor
Another furious Clacton resident raged: “A lot of people have voted for him and now he has decided he’s not going to be that person we voted for.
“He is not going to back us, so I think maybe we should have somebody else.”
A third 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?”
One Clacton resident said Mr Carswell 'no longer represented' residents after he quit the party
There was, however, some support for the now independent MP as one man told the local radio station: “If he wants to stand as an independent I think he’s a good MP. As he’s not standing for another party I don’t see why there should be [a by-election].”
Mr Carswell has represented Clacton since the Essex constituency was formed back in 2010.
In the 2015 General Election, he held on to his seat under the Ukip rosette with 44.4 per cent of Clacton voters backing him.
Nigel Farage in pictures Thu, December 15, 2016
Nigel Farage is a British politician who has been the leader of the UK Independence Party since October 2016.
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British politician Nigel Farage arrives at Trump Tower on December 15, 2016 in New York City. President-elect Donald Trump and his transition team are in the process of filling cabinet and other high level positions for the new administration.
The 45-year-old had previously had a tetchy relationship with some senior Ukip figures, including one the party’s biggest donors, Arron Banks, who described his departure as a “rare piece of good news for Ukip”.
“Carswell has been nothing but a source of division for the party since he joined,” Mr Banks said.
“A point which Nigel [Farage] and I have stressed constantly for months now.
"His resignation is a good first step, but Ukip will remain in a state of constant dysfunction until the rest of Carswell's Tory cabal follows him out of the door."