The issue is a matter for "Scots living in Scotland" and should not become a glorified general election, David Cameron wrote in a letter to Alex Salmond.
Any TV debate should instead involve Alistair Darling, the Labour MP leading the Better Together campaign to keep Scotland in the United Kingdom, he claimed.
"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," the Tory leader wrote.
"I understand why you might wish to pursue a diversionary tactic. It is a convenient means of 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.
"You want the independence debate to be an argument between you and me; the Scottish Government and UK Government; the SNP and Conservative Party - in fact anything rather than what it really is about. Nor is your argument with the rest of the United Kingdom, it is with the people in Scotland."
The consequences of the vote on September 18 next year will be "irreversible" and be felt long after both politicians have retired from politics, he continued.
"The referendum is therefore too important to be reduced to the status of some glorified general election," he wrote.
"People should cast their vote in the knowledge that 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."
It follows repeated invitations from Mr Salmond who insists the Prime Minister is the most senior politician arguing for a No vote.
Mr Salmond said the UK Government makes decisions affecting Scotland, so Mr Cameron should take part in a debate.
It would give him an opportunity to counter the "spurious" claims about an independent Scotland, he said.
The First Minister said: "The highly political nature of the Prime Minister's letter rather makes my point for me. He is in the impossible position of continuing to enter the debate on Scottish independence without actually being willing to have a head-to-head debate.
"I would like the opportunity to counter the various spurious and unfounded claims about an independent Scotland he has made in his letter, and the best way to do that is by way of a live televised 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. 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.
"I have noted the Prime Minister's apparent unwillingness to take part in another general election debate and I'm sure people will draw their own conclusions from that. Indeed, I believe his refusal to debate Scotland's future with me can be summed up in one word - fear."
A spokesman for pro-independence group Yes Scotland said: "Given the importance of the decision to be taken at next September's referendum, it seems to us entirely appropriate for the First Minister of Scotland and the Prime Minister of the UK to debate the issues surrounding Scotland's future.
"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."
A spokesman for Better Together said: "David Cameron does not have a vote in this referendum, Alistair Darling does. He is ready to debate Alex Salmond any time and any place. The people of Scotland want details about the impact of the SNP's plans for independence on our jobs, our pensions and our currency. That is why we have already written to the broadcasters asking them to begin negotiations on these debates.
"There is only one conclusion to be drawn from Alex Salmond's repeated refusal to debate Alistair Darling. Alex Salmond is running scared."