Mr Farage stood down on Friday, having failed to win Kent's Thanet South seat.
Mr Carswell, who held the Clacton seat he retained in a 2014 by-election after defecting from the Conservatives, said there were more suitable replacements.
"I've ruled myself out... because I can think of at least half a dozen people who could do a better job," the Essex MP said.
Mr Farage had recommended Suzanne Evans, the deputy chairman, as stand-in leader until the leadership challenge is complete. And he left the door ajar for his own return, saying: "I will consider over the course of this summer whether to put my name forward to do that job again."
Mr Carswell told the Sunday Politics that his former boss had "earned the right to have a bit of time" to think about standing for re-election in September.
But he insisted the party was not a "one-man show", adding: "You have got a choice between Patrick O'Flynn, Suzanne Evans, Steven Woolfe, Paul Nuttall. Let's see what they've got to offer in terms of tone, in terms of style and approach.
"We need to listen to what the other candidates have to say and I think we need to take a considered decision in September."
UKIP's only MP, Douglas Carswell, has ruled himself out of the running to replace Nigel Farage as leader.