Former Cabinet minister Ken Clarke said he was "disappointed" by Mr Cameron's decision but said the "chickens came home to roost" after the referendum.
He told BBC News: "I'm a friend of David's, so I regret to say, I think for history his legacy will be he's the man who accidentally caused Britain to leave the European Union.
"There's no getting away from the fact that that will overshadow every other feature of his premiership."
Labour leadership contender Owen Smith said: "David Cameron's legacy will be a country left reeling after six years of failed Tory austerity."
Former No 10 communications director Sir Craig Oliver said Mr Cameron did not want to be seen as a "back seat driver".
"He doesn't want to be the person that people go to cause disagreement or have disagreements with the Government."
"For him duty and loyalty and wanting the Conservative Party to succeed, wanting the Conservative prime minister to succeed, wanting Britain to succeed is very important to him and he doesn't want any sense in which he is standing in the way of that or appears to have a gripe or a problem about that."
