The First Minister, meanwhile, has been accused of "running scared" for issuing the same refusal to debate independence with former Labour Chancellor Alistair Darling, who leads the No campaign.
Last week, The Herald reported how Mr Salmond had made his most bullish attempt yet to secure a live head-to-head debate with Mr Cameron on St Andrews Day.
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In a letter to Downing Street, the First Minister stressed how the Prime Minister and the UK Government were spearheading the No campaign through anti-independence analysis papers and so it was only right for the men to debate the issues on television.
Days later, The Herald revealed how senior Whitehall sources had made it clear a 'PM-FM' debate was a non-starter.
Yesterday, Mr Cameron wrote back to the SNP leader, accusing him of trying to use a live head-to-head between them as a distraction and urged him to take part in one with Mr Darling.
The Prime Minister told Mr Salmond: "It is entirely right for you to place yourself at the head of the Yes campaign but not to decide who should lead for the No campaign too. It is a well understood and reasonable principle that you get to pick your own team's captain but not your opponent's as well."
He accused the First Minister of pursuing a diversionary tactic aimed at deflecting attention away from the real issues: "The lack of credibility of your plans for a currency union, funding pensions and managing volatile oil revenues".
Mr Cameron said the First Minister wanted the independence debate to be an argument between "you and me, the Scottish Government and UK Government, the SNP and Conservative Party" but that it was, in fact, between Mr Salmond and the people of Scotland.
Stressing how any decision to split from the rest of the UK would be felt long after both of them had retired from politics, the PM insisted the referendum was too important to be reduced to the status of a General Election.
"People should cast their vote in the knowledge they are deciding not just for themselves but also for their children, grandchildren and succeeding generations.
"It is for people in Scotland to decide. And it is right for you and Alistair Darling, as the leaders of the respective campaigns with votes to cast as well as votes to win, to debate head-to-head on TV."
But the Scottish Government branded Mr Cameron a "big fearty" while Mr Salmond claimed the highly political nature of the Prime Minister's letter "rather makes my point for me".
The First Minister said the Tory leader was in the impossible position of continually entering the independence debate without being willing to have a head-to-head debate.
"The Government in which Mr Cameron serves as Prime Minister is central to the entire referendum debate from the perspective of the No campaign," said Mr Salmond.
"The reality is his Government continues to make decisions affecting Scotland such as the implementation of the hated bedroom tax and the deeply unpopular privatisation of the Royal Mail, despite the fact an overwhelming majority of Scots didn't vote for him or the Tory Party."
Noting how the PM was displaying an apparent unwillingness to take part in another General Election debate, the First Minister said: "His refusal to debate Scotland's future with me can be summed up in one word, feart."
A spokesman for Better Together, pointing out that Mr Darling had offered to debate with Mr Salmond any time, anywhere, said: "There is only one conclusion to be drawn from Alex Salmond's repeated refusal to debate Alistair Darling. Alex Salmond is running scared."
A spokesman for Yes Scotland added: "We favour a full and frank debate because we have absolute confidence that the case for Yes will win through. Mr Cameron's refusal to debate with Alex Salmond would suggest that the Prime Minister might agree with us."