The Special Olympics proposal did not appear serious from the start. The day before it was released, the White House announced that Vice President Mike Pence’s wife, Karen Pence, would lead the American delegation to the games.

That did not stop the friction between Ms. DeVos, congressional Democrats and several Republicans.

Ms. DeVos had said that the program, which received an appropriation of about $17.5 million, did good work but that in a time of difficult budget decisions, hard choices had to be made.

Matt Wolking, deputy communications director of Mr. Trump’s re-election campaign, said Thursday on Twitter that “the Special Olympics, by the way, is a private charity with $61.3 million in net assets that gets only 10% of its funding from taxpayers and enjoyed a $1.8 million surplus in FY2016.” He also said, “I’m sure that Democrats who see abortion as the cure for Down syndrome and other disabilities are sincerely concerned about kids having a chance to be in the Special Olympics.”

That sentiment prevailed for about four hours until Mr. Trump quickly reversed course.

“I want to fund the Special Olympics, and I just authorized a funding of the Special Olympics,” Mr. Trump said. “I’ve been to the Special Olympics. I think it’s incredible, and I just authorized a funding. I heard about it this morning. I have overridden my people. We’re funding the Special Olympics.”

Ms. DeVos also noted that she had long been a supporter of the Special Olympics. A billionaire, she received approval to give away her $199,700 government salary last year, and part of it went to the organization.