THE GROUP OF 20 summit in the French Mediterranean resort of Cannes has ended amid continued uncertainty over Europe’s financial crisis — and hand-wringing in some quarters about the United States’ supposed inability to influence the situation. With debt problems of its own, the story goes, the United States can do little but stand by and sweat as Europe’s woes threaten U.S. prosperity. Only cash-rich China is in a position to aid Europe — and exploit the situation for its own advantage.

There’s some truth to this. Obviously a fiscally sound America is a stronger America, both at home and abroad. If the United States were flush, it might find it easier to organize an international rescue through the International Monetary Fund, where the United States still carries considerable sway, without tapping China. When President Obama or Treasury Secretary Timothy F. Geithner have undertaken to advise the Europeans on how to get their house in order, the response has sometimes been “physician, heal thyself.”

Still, the message Mr. Obama and most of his G-20 colleagues sent the Europeans at Cannes — fix this problem yourselves — was not a reflection of U.S. weakness but of the United States’ best interests and the best interests of Europe. Even if we had the money to rescue the euro, it’s not clear that we should make such an investment, unless and until Europe itself had exhausted its resources, which it has not yet done. And if Europe’s leaders are serious about punching their weight in foreign affairs, they can do that only on the basis of economic self-sufficiency.

Surely the European project should not be mortgaged to communist-ruled China. The only reasons China might want to bail out Europe are self-interested ones, such as maintaining the euro as a counterweight to the dollar in global currency markets, or winning renewed European arms sales and indulgence on human rights issues.

Alas, the prospects that Europe will resolve its problems are not much clearer after Cannes than they were before. In a moment of unintentional comedy Friday, President Obama ticked off all the European institutions that have to be consulted before anything gets done, concluding with the understatement that, “trying to coordinate all those different interests is laborious.” The eurozone as presently constituted may, in the end, be ungovernable.

But there are a couple of hopeful signs. Italy’s Prime Minister Silvio Berlusconi, after weeks of failing or refusing to enact irreversible structural reforms, has at least agreed to submit his country to quarterly monitoring by the IMF. In Greece, Prime Minister George Papandreou has backed off his disastrous plan for a referendum on membership in the euro, and the country’s political class appears more united on the basic necessity of reform in return for debt relief.

What remains, as Mr. Obama noted repeatedly, is for Europe’s major powers, Germany especially, to put together a pool of capital large enough to guarantee Italy’s debt — and thus to deter further speculation against it. Yes, this is easy for Mr. Obama to say, and hard for the Europeans to do. But if the Europeans mean it when they say that the fate of their union itself depends on saving the euro, they will find a way.