Washington (CNN) President Barack Obama's administration admitted it erred by failing to send a higher-ranking representative of the United States to the Paris unity march on Sunday.

"I think it's fair to say that we should have sent someone with a higher profile to be there," White House Press Secretary Josh Earnest said Monday afternoon.

He said Obama himself would have liked to attend the march "had the circumstances been a little different." But planning began Friday night, 36 hours before the event began, and there wasn't enough time for the "onerous and significant" security work that needed to take place ahead of a presidential visit, Earnest said. He said Obama's presence also would have meant extra restrictions on the people who were there.

"That said, there is no doubt that the American people and this administration stand foursquare behind our allies in France as they face down this threat," he said. "And that was evident throughout last week."

More than 40 world leaders, including the British, German and Israeli heads of state and Russia's foreign minister, joined at least 1.5 million people on the Paris streets Sunday for a unity march that became France's biggest-ever public demonstration.

But Obama and his administration's top hands were nowhere to be found -- an absence that triggered complaints that he missed a key leadership opportunity.

The United States appeared to have options to send to the march: Obama spent Sunday at the White House with no public events on his schedule. Vice President Joe Biden was at home in Delaware for the weekend, also with a blank public schedule. Outgoing Attorney General Eric Holder was already in Paris for security meetings -- and even recorded interviews with several U.S. Sunday morning programs -- but he didn't attend the march.

A Secret Service official said the agency was not asked to draw up security plans for a potential presidential trip to Paris in advance of Sunday's march.

"We weren't asked or notified about a trip," the official said. But the agency had Secret Service agents on the ground in Paris, per its standard operating procedure.

"It would have been a challenging advance ... based on what we know," Secret Service spokesman Brian Leary said. But Leary did not say that such an advance would have been impossible.

During the White House briefing, Earnest suggested security challenges were a factor in not having the president travel to Paris. But Earnest acknowledged the Secret Service could have pulled it off. An agency official noted previous "last minute" presidential trips have happened during the Obama presidency, including a hurried visit to South Africa in December 2013 for the memorial service for Nelson Mandela.

The White House noted that it was represented in Paris on Sunday -- and has offered support to France in recent days.

U.S. Ambassador to France Jane Hartley was in the march, as was assistant Secretary of State Victoria Nuland. At a security summit, Holder was joined in those security meetings by deputy Homeland Security Secretary Alejandro Mayorkas.

Obama personally visited the French Embassy in Washington last week to offer his support.

Secretary of State John Kerry, meanwhile, will visit Paris on Friday.

Kerry skipped Monday's march because he was in India on Monday for a long-planned event there with new Prime Minister Narendra Modi -- a key relationship as the United States tries to improve long-strained trade ties with the country.

Kerry brushed the criticism off as "quibbling," saying he'll visit Paris on his way back to the United States to make "crystal clear how passionately we feel" about the attacks and response.

"The U.S. has been deeply engaged with the people of France since this incident occurred," Kerry told reporters, adding that the United States has offered intelligence and law enforcement help.

"This is sort of quibbling a little bit in the sense that our assistant Secretary of State Victoria Nuland was there and marched, our ambassador was there and marched, many people from the embassy were there and marched."

France's ambassador to the United States, Gerard Araud, sought to show there are no hard feelings, tweeting on Monday: "I am extremely grateful for the overwhelming support France has received from everybody here, from the President to the ordinary American."

The White House's push-back comes as Obama takes heat -- particularly from Republicans considering 2016 presidential bids -- for his absence.

Rick Perry tweeted that Obama "should have stood with France in person to defend Western values and show support for victims."

Sen. Ted Cruz (R-Texas) wrote for Time: "Our President should have been there, because we must never hesitate to stand with our allies."

And Sen. Marco Rubio (R-Fla.) said Monday that "it was a mistake not to send someone."

Rubio said on CBS' "This Morning" that he understands that the President's security detail can be problem in mass gatherings like the rally, but suggested Holder or Kerry should have gone in his place.

"I think in hindsight, I would hope, that they would do it differently," Rubio said.

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Who did go

British Prime Minister David Cameron, German Chancellor Angela Merkel, Israeli Prime Minister Benjamin Netanyahu, Palestinian Authority President Mahmoud Abbas and Russian Foreign Minister Sergey Lavrov were among those who attended, along with religious leaders.

Fareed Zakaria, host of CNN's "Global Public Square," called the absence of top U.S. officials a mistake.

France is the United States' "deepest ideological ally," he said, and it would have been a meaningful image to have a senior administration member, or the President, standing shoulder to shoulder with other leaders.

Zakaria noted that security concerns didn't dissuade Netanyahu or Abbas or other leaders from showing up. But Obama's absence did show that the struggle against radical Islam is "not all about America," Zakaria said.

"Many people have tended to think that Islamic terrorism wouldn't exist without America," Zakaria said. "This is really a struggle between the civilized world and a band of extremists. Even if you take the U.S. out of it ... the civilized world is up in arms."

And Jake Tapper, host of CNN's "The Lead," said American leaders were conspicuously absent from historic Paris rally, perhaps the most important public demonstrations in Europe in the last generation.

Tapper not only called out the President and his administration, but also prospective 2016 hopefuls from both parties, for missing the opportunity to share in the global moment.

What Obama said

Obama spoke about the Paris attacks on Friday, saying he wants the people of France to know the United States "stands with you today, stands with you tomorrow."

The White House also announced Sunday that it will host a February 18 summit aimed at countering violent extremism.

Earnest said that event will "highlight domestic and international efforts to prevent violent extremists and their supporters from radicalizing, recruiting or inspiring individuals or groups in the United States and abroad to commit acts of violence, efforts made even more imperative in light of recent, tragic attacks in Ottawa, Sydney and Paris."

He said the summit will include presentations, panel discussions and small group meetings, focused on the local, state and federal government levels.