Russia’s aggression in Ukraine is a “wake-up call” for the Atlantic military alliance and other international institutions that have buttressed European security and stability for decades, NATO Secretary General Anders Fogh Rasmussen said Wednesday.

“We live in a different world than we did less than a month ago,” Rasmussen said in a previously scheduled Brookings Institution speech that was adjusted to reflect a sudden crisis that he called Europe’s “gravest threat . . . since the end of the Cold War.”

How NATO and its 28 individual members respond to the new world Rasmussen outlined is likely to determine whether the challenges that have plagued the alliance almost since its inception are eased or aggravated.

Many of those challenges — American dominance, unequal burden-sharing, defense budget woes — were somewhat subdued during NATO’s first four decades, when the United States was ready and eager to lead on the front lines of the Cold War.

But since the 1991 demise of the Soviet Union, with interruptions for joint action in places such as Bosnia, Afghanistan and Libya, those problems have dominated virtually every high-level NATO meeting.

President Obama, who told a San Diego television station Wednesday that “we are not going to be getting into a military excursion in Ukraine,” came to office with a commitment to build stronger international institutions. But all three of his defense secretaries, from Robert M. Gates to Leon Panetta to Chuck Hagel, have alternately berated and pleaded with Europe to increase defense spending.

“America’s contributions in NATO remain starkly disproportionate, so adjustments in the U.S. defense budget cannot become an excuse for further cuts” in Europe, Hagel told his European counterparts in a closed-door session last month in Brussels just after he announced new decreases in U.S. spending.

The strategic plan that NATO adopted in 2010 called for increased spending on capabilities, including cyberwarfare, intelligence and surveillance. Yet in 2013, only a handful of NATO countries other than the United States, including Estonia, Greece and Britain, achieved the alliance’s defense spending goal of 2 percent of gross domestic product, and many weren’t even close.

Britain and France, which reached 1.9 percent in 2013 according to NATO figures, have announced major cuts this year, leading to new charges from American critics that Europe continues to seek a “free ride” from the United States.

“I am the first to stress that Europe must do more,” Rasmussen said. “Developments in Ukraine are a stark reminder that security in Europe cannot be taken for granted.”

At NATO’s upcoming September summit in Britain, he said, “we need to take tough decisions” to demonstrate that “our commitment to the security of allies is unbreakable.”

At the same time, Rasmussen argued, NATO is not entirely a one-way street that the United States has built, paved and continues to maintain.

“Over the past 10 years, for every two U.S. soldiers who have served in Afghanistan, one European soldier has always served with them,” totaling 400,000, he said.

NATO countries continue to keep the peace in Kosovo, and in NATO’s 2011 Libya operation, “European allies, Canada and NATO partners played a crucial role in enforcing an arms embargo, maintaining a no-fly zone and protecting the people from attacks by their own leader,” Rasmussen said.

“It comes down to a simple truth: Shared security is better than solitary insecurity. And it’s cheaper, too,” he said. NATO, he added, is “a great deal for America. And it’s why NATO matters to America.”

Others have argued that some of those missions provided as much evidence of NATO dysfunction as they did of equality and cooperation. Although the United States quickly pulled back to let others take the lead in Libya, the operation revealed significant shortfalls in European capabilities. Although NATO agreed by consensus to participate, a number of countries declined, leading to charges that the alliance had become an “a la carte” organization.

In Afghanistan, caveats by individual nations on what kinds of combat and other missions they were prepared to undertake led to significant U.S. frustration.

“The irony is that the alliance has been focusing a lot in the last decade on threats outside of Europe,” said Stephen Larrabee, an European security expert at the Rand Corporation.

The question now is whether a return to a more direct threat to Europe, “right on NATO’s border,” as Rasmussen put it, will focus minds and open pocketbooks.

“This is a time where it ought to be a little easier for parliaments and government to make the case that a decade and a half of underinvestment in defense needs to come to an end,” said Ivo Daalder, president of the Chicago Council on Global Affairs, who served as Obama’s ambassador to NATO for four years.

“I would hope that we spend a lot of time making clear that’s what we expect them to do,” Daalder said. “We’ve done that for many years and gotten nowhere.”

Larrabee was pessimistic. “There will be a lot of talk,” he said, “but I’d be surprised if you see much action.”

Ukraine has also raised new questions about NATO’s massive expansion after the fall of the Soviet Union. After a long debate in the early 1990s about moving eastward, the decade between 1999 and 2009 brought 12 new countries aboard.

Four others — Macedonia, Bosnia and Herzegovina, Montenegro, and Georgia — have expressed interest in joining.

Retired Adm. James G. Stavridis, who served as supreme allied commander of NATO from 2009 until 2013, said the crisis is likely to have a “salutary effect” on financial commitments NATO members have made but failed to keep. Overall, he said, Russia’s actions in Ukraine stand to boost the alliance’s relevance and jolt its politics, especially among its newer members in the east “who quite vividly remember being under the Russians. They’re nervous,” he said.

During a visit this week from Vice President Biden, however, Baltic leaders seemed to rest their hopes squarely on the United States and praised what Latvian President Andris Berzins called Biden’s “unwavering reassurance.”

Ernesto Londoño contributed to this report.