While politicians from both parties concoct situations for political gain, some criticized Mr. Pence’s walkout as transparently premeditated. The vice president did not take a pool reporter traveling with him into the stadium; a member of Mr. Pence’s staff told the reporter, Vaughn Hillyard, that the vice president might be leaving the game early.
“Manipulation of faux patriotism took new turn today with VP Pence. Preplanned early exit from Colts game after 49ers kneeled, then tweets,” Norman Ornstein, a resident scholar at the conservative American Enterprise Institute, wrote on Twitter.
Others pointed out the expense involved: Mr. Pence flew to Indianapolis from Las Vegas, where he had attended a memorial service for victims of last Sunday’s mass shooting, and was immediately flying back to Los Angeles.
“After all the scandals involving unnecessarily expensive travel by cabinet secretaries, how much taxpayer money was wasted on this stunt?” Representative Adam B. Schiff, Democrat of California, said in a tweet.
There was little doubt, given the presence of the 49ers, that Mr. Pence would be given an opportunity to make his political statement. The former 49ers quarterback Colin Kaepernick began the dispute over the national anthem last year by taking a knee to highlight the plight of black Americans, particularly the killing of black men by police officers.
Mr. Kaepernick left the 49ers in March and has not been signed by any other team — a situation seen by many as a blacklisting by other team owners. But other 49ers have continued the protest in a show of solidarity with their former teammate.
Last month, Mr. Trump seized on the dispute over Mr. Kaepernick, declaring at a rally in Alabama that any “son of a bitch” who refused to stand for the anthem should be “fired.” His comments led hundreds of players to protest during the anthem, though last weekend the furor appeared to be subsiding a bit.
The White House seemed determined to whip it up again. In addition to Mr. Pence’s statement and tweets, the press office released a photo of Mr. Pence in a suit, standing with his hand over his heart for the playing of “The Star-Spangled Banner” at Lucas Oil Stadium in Indianapolis. He was standing next to Maj. Gen. Courtney P. Carr, a veteran of the Iraq war who now serves as adjutant general of the Indiana National Guard.
An earlier tweet by the vice president, in which he said that he and his wife, Karen Pence, were looking forward to the game, showed him wearing a Colts cap and casual clothing. He said he was there to honor Peyton Manning, the longtime Colts quarterback whose statue was unveiled in Indianapolis this weekend, and who was set to have his number retired into the team’s ring of honor at halftime.
The vice president’s office later said that the photo had actually been taken in 2014, when Mr. Pence was governor of Indiana.
In the West Wing, officials privately said they believed that Mr. Pence had sought the controversy, which is a favorite of the president’s and which he had voiced his approval of. It came less than two weeks after Mr. Pence’s chief of staff, Nick Ayers, told a group of Republican donors that there might need to be a “purge” of the party if people did not get behind the president.
Two weeks ago, Mr. Trump had called for fans to walk out of stadiums if players knelt. So Mr. Pence was doing what Mr. Trump had called for, though few fans seem to have followed his call.
Mr. Pence has found himself on the other side of this kind of protest. Shortly after the election, he attended a Broadway performance of “Hamilton,” the hit musical about the founding fathers, and received a polite lecture from a member of the cast, after the curtain call, on the need for the Trump administration to work on behalf of all Americans.
“We truly hope that this show has inspired you to uphold our American values,” Brandon Victor Dixon, the actor playing Vice President Aaron Burr, said from the stage.
Mr. Pence seemed unperturbed, but Mr. Trump reacted angrily, saying the cast had “harassed” the vice president-elect and owed him an apology.
In the weeks since Mr. Trump complained about players kneeling during the anthem, most have returned to standing, though some continue to kneel, particularly on the 49ers. While the 49ers knelt, the Colts locked arms as a team on Sunday.
In Detroit, the Lions stood arm in arm, including two players who had knelt the past two weeks. In New Jersey, the Giants defensive end Olivier Vernon, who was in street clothes, knelt while the rest of his team linked arms.
In Miami, three members of the Dolphins — Kenny Stills, Julius Thomas and Mike Thomas — stayed in the locker room while the rest of their team stood on the sideline during the anthem.
The Dolphins’ owner, Stephen M. Ross, who has backed the players’ right to protest, appeared to be changing his stance. He told The Miami Herald that while many players insisted that their protests were about raising awareness of social injustice, the president had “changed that whole paradigm of what protest is” by turning it into a proxy for respect for the flag and support of the military.
Mr. Kaepernick, meanwhile, has been working out privately in New Jersey in hopes of finding a team willing to take a chance on him, according to a CBS report.
Mr. Pence framed the issue as one of patriotism and respect.
“At a time when so many Americans are inspiring our nation with their courage, resolve and resilience, now, more than ever, we should rally around our flag and everything that unites us,” he said in a statement. “While everyone is entitled to their own opinions, I don’t think it’s too much to ask N.F.L. players to respect the flag and our national anthem.”