By Ian Talley 

The Obama administration, after pressing Greece and its European creditors to compromise for weeks, returned to its familiar role on Monday of prodding from the sidelines.

A day after Greek voters soundly rejected bailout terms offered by its creditors, the White House renewed its calls for Athens to restructure its economy and for Europe to offer the beleaguered county hope by offering better terms on its debt.

Keeping Greece in the eurozone "will require both a package of financing and reforms that will allow Greece to achieve, or at least be on a path towards, some debt sustainability," White House press secretary Josh Earnest said.

"But it is also important for Greece to implement the kinds of reforms and to keep the commitments that they've made previously," he said.

The calls from the world's most powerful country may do little to change the fate of negotiations. The U.S. is much more of a bystander in the latest stage of the European crisis, struggling to sway the deeply entrenched sides.

"They're marginalized," said Robert Kahn, a senior fellow at the Council on Foreign Relations and former economist at the International Monetary Fund. Compared with 2010, when the U.S. was front and central to resolving the global financial crisis, "people just don't listen as much to us."

For months, U.S. officials have pushed Greece and its creditors toward what they called a "pragmatic compromise." If Greece would vow to tough economic overhauls and its European creditors would consider debt relief, U.S. officials argued, Athens could avoid falling deeper into economic crisis and the eurozone could prevent further hiccups in its long, slow recovery from the financial crisis.

At the same time, Mr. Earnest also played down the potential impact of the Greek crisis on the U.S., saying the U.S. has little direct exposure to the country. Still, he warned that if Greece's woes weighed on the eurozone's economy, it could damp U.S. exports.

Neither Greece nor the European creditors gave enough ground in recent weeks to ease the standoff, leaving U.S. officials to do little more than monitor the fallout.

Now, Treasury Department and Federal Reserve officials will be watching for signs of the crisis spreading throughout Europe and global financial markets. Europe's troubles are just one of area of global weakness the central bank is watching as it considers when to raise interest rates for the first time in nearly a decade.

The U.S. has tried to play the role of independent adviser to both Greece and its eurozone creditors. But Washington's financial diplomacy hasn't yielded much fruit. That is partly because the U.S. has had little leverage to wield on the issue.

Given the eurozone's stouter financial defenses, a healthier U.S. economy and a host of other geopolitical priorities--including Iran nuclear talks, the rise of Islamic State and growing tensions with China--the Greek crisis hasn't been the administration's highest priority.

"We've made clear that we continue to believe it's in the U.S. interest and the global interest for these differences to be resolved, but ultimately, it will be the responsibility of the Europeans to resolve them," Mr. Earnest said.

The eurozone has built up buffers to insulate itself and the global economy from a major disruption in markets. U.S. officials point to the currency union's bailout funds, the European Central Bank's monetary firepower and efforts to shore up the financial system as largely sufficient to ensure the region is much better prepared to weather a collapse of the Greek economy or an exit from the currency union than in previous years.

"You cannot keep on squeezing countries that are in the midst of depression," Mr. Obama said at the time. "At some point there has to be a growth strategy in order for them to pay off their debts to eliminate some of their deficits."

The newly elected Greek prime minister, Alexis Tsipras, took those comments as support for his push to rewrite the terms of his country's bailout with less budget belt-tightening and creditor cuts in Greece's debt burden.

More recently, the administration has largely backed Greece's creditors, the eurozone lenders and the International Monetary Fund.

In dozens of phone calls in the weeks preceding the referendum, U.S. Treasury Secretary Jacob Lew and his lieutenants told top European officials Athens needs to commit to tough economic overhauls while Europe should be prepared to offer Greece debt relief.

Write to Ian Talley at ian.talley@wsj.com