“If the new arrangement turns out to be too toothless to enforce the rules, we’ll be back to square one,” said Thomas Klau, a political analyst and head of the Paris office of the European Council on Foreign Relations.

Just ahead of Mrs. Merkel’s unexpectedly robust success, Mr. Obama issued his unheeded warning from across the Atlantic. “There’s a short-term crisis that has to be resolved,” he said, “to make sure that markets have confidence that Europe stands behind the euro.”

Mr. Obama is fiercely proud of the record he achieved in keeping not just the United States but also the entire world out of an acute financial meltdown after 2008, presiding over enormous stimulus spending in tandem with unrestrained support from the Federal Reserve. The president and his allies now say that in doing so, they may well have prevented the world from falling into another Great Depression.

By ignoring the short-term threat, American officials say, Mrs. Merkel is unwittingly courting the very threat they so narrowly managed to keep at bay. Strong governments can borrow cheaply, mainstream economists on both sides of the Atlantic argue, and have an obligation to intervene more aggressively than they would normally to make up for the slump in private demand.

Germans are staunchly opposed to any solution that involves greater debt, but even more so to policies that might court inflation, their historic obsession. Policy makers in Berlin and at the Bundesbank headquarters in Frankfurt have urged restraint on the part of the European Central Bank, insisting it should not buy up too many bonds from heavily indebted euro zone countries.

“We will save the euro. We have to save the euro. We have the biggest resources and the biggest interest. But we will hearken fiscal probity,” said Josef Joffe, publisher of the German weekly Die Zeit, describing the German position. “We will not sacrifice our memories.”

The Obama team argues that with recession on the horizon for Europe, the threat of inflation is low and the real threat is a great depression. Administration officials, and many economists, argue that Germany is remembering the wrong crisis — it should be focused on a repeat of the deflation and contraction of the 1930s, not a repeat of the hyperinflation of the ’20s.