The Prime Minister attacked Penny Mordaunt’s claims that Britain had no veto over Turkish accession to the 28-member bloc as “misleading” and “absolutely wrong”.
Turkey is set to join the EU in the next eight years and the UK will be unable to block the move, the Armed Forces Minister had insisted.
But Mr Cameron told ITV’s Peston on Sunday: “The Leave campaign are making a very misleading claim.”
Asked if Ms Mordaunt had been wrong to say Britain would not be able to veto its membership, he replied: “Absolutely wrong. Let me be clear, Britain and every other country in the European Union has a veto on another country joining. That is a fact, and the fact that the Leave campaign are getting things as straightforward as this wrong should call in to question their whole judgment in making the bigger argument about leaving the EU.”
Pressed on whether she was qualified to remain in government, the prime minister told the programme: “Her responsibilities are in the Ministry of Defence, she is doing a very good job.
“But on this question of whether of not we have a veto, the Leave campaign are wrong.”
Mr Cameron added: “It is not remotely on the cards that Turkey is going to join the EU any time soon. They applied in 1987. At the current rate of progress they will probably get round to joining in about the year 3000 according to the latest forecasts.”
As the increasingly bitter clashes between the Remain and Leave camps fuelled divisions within the Conservatives, Mr Cameron insisted the party would “come back together”.
“Of course this issue raises great passions. It is not surprising that you have people in a political party on either sides of the arguments,” he said.
“I’m absolutely convinced that at the end of this the Government, the Conservative Party, will come back together and get on with the important job of running the country.”
Asked if Boris Johnson’s comparison between Adolf Hitler and the EU had made it impossible to give him a job in government, the Prime Minister told the programme: “I’m not going to get into jobs and the future now.”
“Boris is hugely capable in lots of ways but I’m not going to go into Boris today,” he added.
Mr Cameron insisted that he plans to serve a “full second term” in No 10.
Asked if that means he plans to stay in his post until 2019, he said: “It means a full second term, it means what it says.”
The premier sidestepped questions about Chancellor George Osborne’s chances of succeeding him, telling the programme: “I think he’s a man of great talent.
“I’m not going to pick my, luckily I don’t have to, pick my successor, the Conservative Party will do that.
“But George is hugely talented, he’s a brilliant Chancellor of the Exchequer.
“You can see it in the economy; 2.5.million more people in work, almost a million more businesses since he and I got our jobs, the economy growing and so I think he’s a great talent.”
He added: “I think in politics you have to have partnerships. Prime ministers do not do these jobs on their own.”
Leave campaigners claim Britain will open the doors to high levels of murderers, terrorists and kidnappers from countries like Turkey if it remains in the EU.
Turkish membership will also cost NHS maternity services £400 million in a decade, they warned.
Ms Mordaunt said it is “very likely” that Turkey will join the EU in the next eight years, and insisted Britain “doesn’t” have a veto to stop such a move.
She told BBC One’s Andrew Marr Show: “I do not think the EU is going to keep Turkey out. I think it is going to join, I think the migrant crisis is pushing it more that way.”
Pressed on her veto claims, Ms Mordaunt said that in the face of the migrant crisis “we will be unable to stop Turkey joining”.
“This is a matter for the British people to decide and the only shot that they will get at expressing a view on this is in this referendum.
“I don’t think the UK will be able to stop Turkey joining.”
Ms Mordaunt denied the warnings about Turkey were “dog-whistle” claims, telling the programme: “Those facts and figures are agreed on by both sides of this debate.”
The minister accused the Remain camp of “scaremongering” and “false reports”.
“I think that the public have got tired of this constant stream of hysteria from the establishment,” she said.
Ms Mordaunt said Bank of England governor Mark Carney’s warning that quitting the EU could spark a recession “was the tipping point when people realised there is another agenda going”.
Asked if the In campaign was an “establishment stitch-up”, she replied: “I think so.”
James McGrory, chief campaign spokesman for Britain Stronger in Europe, accused the minister of “lying”.
He said: “Mordaunt is plain and simple lying to people. We have a veto on whether Turkey can join the EU. As does every other EU nation.
“That was a disgraceful from Penny Mordaunt on @MarrShow. Criticises dishonesty and then is utterly dishonest about our veto on Turkey.”
In 2010, Mr Cameron told Turkey he would “remain your strongest possible advocate for EU membership and for greater influence at the top table of European diplomacy”.
Ukip leader Nigel Farage said: “Veto? Mr. Cameron has said he wants to pave the road from Ankara to Brussels.”
The row came the day before the one-month countdown to the referendum on June 23.