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.