The school’s website and Facebook page were down as of Saturday afternoon.

The encounter became the latest touchpoint for racial tensions in America, particularly under Mr. Trump, who has painted immigrants in broad strokes as rapists and drug dealers and recently mocked Senator Elizabeth Warren with a reference to Wounded Knee and Little Bighorn, sacred ground for Native Americans whose ancestors fought and died there.

Across the country, Mr. Trump’s name — and his campaign for a wall on the southern border with Mexico — have been used to goad minorities, including by high school students at sporting events.

[Read more about the Wounded Knee massacre, the Battle of Little Bighorn and why the president invoked them to attack Senator Elizabeth Warren, a 2020 presidential candidate. And a new book by David Treuer, “The Heartbeat of Wounded Knee,” shows the history of American Indians as more than victimhood.]

The episode drew widespread condemnation from Native Americans, Catholics and politicians alike.

“This veteran put his life on the line for our country,” Representative Deb Haaland, a Democrat of New Mexico who recently became one of the first Native American women to serve in Congress, said on Twitter. “The students’ display of blatant hate, disrespect, and intolerance is a signal of how common decency has decayed under this administration. Heartbreaking.”

Sisters of Mercy, a group of Roman Catholic women who take vows of poverty, chastity, obedience and service, condemned the behavior in the videos as disturbing and bigoted. “Racism and intolerance in all forms go directly against Catholic social teaching,” the Sisters of Mercy said.

In a statement on Saturday, the Indigenous Peoples Movement identified the man in the videos as Nathan Phillips, an Omaha elder, a veteran and the former director of the Native Youth Alliance, a group that works to ensure that traditional culture and spiritual ways are upheld for future generations. Mr. Phillips also holds an annual ceremony honoring Native American veterans in Arlington National Cemetery, the group said.

Mr. Phillips could not be reached for comment on Saturday. He told The Washington Post that he noticed the teenagers taunting participants at the Indigenous Peoples March.