“Randy Levine is a great friend of mine from the Yankees,” Mr. Trump, referring to the president of the baseball team, told reporters on Thursday as Dr. Fauci was preparing to take the mound. “And he asked me to throw out the first pitch, and I think I’m doing that on Aug. 15 at Yankee Stadium.”

There was one problem: Mr. Trump had not actually been invited on that day by the Yankees, according to one person with knowledge of Mr. Trump’s schedule. His announcement surprised both Yankees officials and the White House staff.

But Mr. Trump had been so annoyed by Dr. Fauci’s turn in the limelight, an official familiar with his reaction said, that he had directed his aides to call Yankees officials and make good on a longtime standing offer from Mr. Levine to throw out an opening pitch. But no date was ever finalized.