The Prime Minister returned from talks in Berlin with the German leader having made little progress in agreeing emergency action to stop the financial contagion spreading.
Tensions were inflamed after a close ally of Ms Merkel predicted Britain would eventually adopt the euro.
The German media joined the clamour, with the mass-circulation newspaper Bild questioning whether it might be better for Britain to leave the European Union altogether.
Behind the leaders' smiles at a joint press conference yesterday, they acknowledged fundamental differences remained on three key issues:
* New eurozone rules. Ms Merkel called for "limited" changes to European treaties to impose fiscal discipline on the single currency but stressed negotiations should only be for eurozone members. Mr Cameron wants Britain involved in the talks because of the potential impact of the decisions on the UK;
* Whether the European Central Bank should intervene to support the eurozone. Ms Merkel – backed by the German public – is fiercely resisting the move, which she fears would fuel inflation. But Mr Cameron insisted that all the eurozone's institutions had to "do what is necessary to defend it";
* Taxing financial transactions within the EU. Ms Merkel supports the step but Mr Cameron fears it would disproportionately hit the City and said it would work only if applied globally.
The Prime Minister said: "It is obvious we don't agree on every aspect of European policy, but I am clear we can address and accommodate and deal with those differences."
He also stressed the two leaders were "very good friends" and "absolutely" in agreement on the importance of completing the single market, budget discipline and stopping EU spending from rising by more than inflation.
But shortly before Ms Merkel also paid tribute to the "strong bonds of friendship" between the countries, her veteran Finance Minister used less diplomatic language in which he seemed to predict the end of sterling.
Wolfgang Schäuble told the news agency DPA it was Britain's right to remain outside the eurozone "for the time being".
But he said it was a matter of time before non-eurozone states became convinced of the euro's advantages. "One day the whole of Europe will have a single currency and perhaps it will happen more quickly than many people on the British island think," he said.
Meanwhile, in an article headlined 'The Sick Empire', Der Spiegel magazine described Britain's plans to eradicate its budget deficit by 2015 as "utopian". It added: "The situation on the island is more dramatic than in parts of the continent. It's bad news nearly every day.
"But the British government gets away with it by proclaiming carry-on-as-usual policies and by blaming its economic stagnation on the eurozone."
The war of words between Berlin and London erupted on Tuesday after Volker Kauder, Ms Merkel's parliamentary party leader, lambasted Britain for being too self-centred on Europe.
"Just looking for their own advantage and not being prepared to contribute – that cannot be the message we accept from the British," he told a congress of his ruling conservatives.
The former Prime Minister, Sir John Major, weighed in behind Mr Cameron last night as he condemned the financial transaction tax as "a heat-seeking missile...aimed at the City of London". He also warned of an "undemocratic" move towards eurozone fiscal union.
In an interview with Al Jazeera, he also predicted "one or two countries" would be forced to quit the euro.
Douglas Alexander, the shadow Foreign Secretary, said: "David Cameron is leaving Berlin just as isolated as he was when he arrived. "That is a genuine concern for those of us who want Britain to be a strong and, indeed, leading voice in trying to find a way forward."
Great Divide: Tensions behind the soundbites
Cameron
On taxing the banks
He said: "The danger, we have always believed, is driving transactions to a jurisdiction where it wouldn't be applied. So a global tax would be a good thing, but in Britain also we have put in place stamp duty on share transactions, a bank levy."
He meant: Unless there is a global agreement on a "Robin Hood tax", then countries without it – probably in the Far East – will be the biggest winners as financial services are likely to head there. And the biggest loser will be the City of London.
On intervention by the European Central Bank
He said: "All the institutions of the eurozone have to stand behind and back and do what is necessary to defend it."
He meant: Most importantly, that includes the European Central Bank. I appreciate the Germans' historical distrust of the move, only massive ECB intervention can calm the markets.
On treaty change
He said: "We had a discussion about the issue of the treaties and, as Angela has said, Germany has her interests and so does Britain."
He meant: There are plenty of hard negotiations ahead. If those of you with the euro intend to make agreements that will affect the rest of us, think again. We want to be involved in the talks over any new rules for the eurozone.
Merkel
On taxing the banks
She said: "We are at one saying that a global financial transaction tax would be implemented by both countries immediately. But just a European one, we did not make any progress. We have to both work on where we feel change is needed."
She meant: We Germans remain committed to a tax on financial transactions, but we're getting nowhere in persuading the British.
On intervention by the European Central Bank
She said: "The British demand that we use a large amount of firepower to win back credibility for the euro zone is right. But we have to take care that we don't pretend to have powers we don't have, because the markets will figure out very quickly that this won't work.
She meant: Even if I wanted to (and I don't) I haven't got the power to authorise the ECB stepping in. My Parliament, not to mention the German electorate, simply wouldn't wear it.
On treaty change
She said: "It is crucial... we develop more of a possibility of enforcing the rules of the European institutions so that national governments do abide by their commitments. A limited treaty change, only for the members of the eurozone, is of the essence."
She meant: I'm sorry, but if you're not in the single currency club, you can hardly take part in drawing up its new rules. However, rest assured we don't want a Lisbon treaty mark 2. A new bout of EU navel-gazing would be in no-one's interests (and risk referendums around the EU).
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