In New York for a UN meeting last night, Mr Cameron said he was mortified to have breached the long-established protocol that content of conversations between the Queen and prime ministers is never discussed.
He said Downing Street officials had already been in touch with Buckingham Palace to offer his apologies, and indicated that he would personally say sorry to Her Majesty when he next has his regular face-to-face audience with her.
But the conversation may have to wait unless he makes it by telephone as they are not due to meet until she returns from Scotland in early October.
Asked if he regretted discussing the Queen's thoughts and whether he would apologise to her, he replied: "Yes and yes."
He added: "Look, I'm very embarrassed by this. I'm extremely sorry about it. It was a private conversation, but clearly a private conversation that I shouldn't have had and won't have again. My office has already been in touch with the palace to make that clear and I will do so as well."
A Buckingham Palace source said: "We are, of course, happy to facilitate any telephone call from the Prime Minister to Her Majesty."
The apology came as Mr Cameron faced intense criticism over his gaffe.
Dr Andrew Smith, teaching fellow of modern history at University College London said: "While it doesn't seem unnatural for the Queen to have an opinion ... it was a particularly sensitive phone call telling her that her kingdom was intact.
"It does seem like Cameron has been terribly indiscreet, as if he was playing up [to] the bigger boys."
He added: "It was very naive not to think there was a chance of the conversation being picked up ... given his choice of language, the [description of her] purring and the fact that there is an important constitutional convention here, I have to say it will be the most awkward phone call of his political career."
Footage emerged on Tuesday showing the Prime Minister laughing and joking with Mr Bloomberg. Although the pair knew they were being filmed, the fact their comments were picked up by microphones appears to have taken them by surprise.
Mr Cameron can be heard to say: "The definition of relief, if you are Prime Minister of the United Kingdom, is ringing up Her Majesty the Queen and saying, "Your Majesty, it is all right, it's OK." That was something. That's relief. She purred down the line."
By longstanding convention, any discussion between the monarch and their prime minister should remain private, enabling the sovereign to 'advise, encourage and warn' with complete confidence.
The fact the Prime Minister revealed details of such a sensitive conversation with the Queen simply to make idle chit chat will likely concern and anger her senior aides.
Leading up to the referendum, Buckingham Palace stressed the Queen's neutrality and firm belief that the vote was a matter for the people of Scotland.
Scottish First Minister Alex Salmond said Mr Cameron should "hang his head in shame".
The former SNP leader, who led the defeated nationalist campaign, gleefully told BBC Scotland: "David Cameron has been prime minister for four years and he hasn't learned basic civility on not gossiping about what Her Majesty the Queen thinks ... That's absolutely pathetic and he should hang his head in shame."
Labour MP Paul Flynn described Mr Cameron yesterday as a 'gossiping motormouth', adding: "I think it's a major blunder ... particularly embarrassing because he was speaking about a sensitive and divisive political issue."
But Professor Robert Hazell, director of the Constitution Unit at University College London said: "[Mr Cameron] had not intended his private remark to Bloomberg to become public. He will be embarrassed ... but no lasting damage has been done."
- Daily Mail
