Mr. Trump’s comments came hours after he called for a day of national prayer in the country.

As passengers returning from Europe faced long waits at airports for health screenings, Mr. Trump initiated a Twitter broadside against Mrs. Clinton, his opponent in the 2016 election.

“Great Job by Judicial Watch,” Mr. Trump said, promoting a news alert from a nonprofit group about a legal development in its continuing fight to obtain Mrs. Clinton’s depositions about her emails as well as records related to the 2012 Benghazi attack. “Potentially a treasure trove,” Mr. Trump wrote. “Too bad you are not given more help, but it will all work out!”

Mr. Trump, who skipped his Sunday routine on the golf course, also sought to suggest the Obama administration played a role in the current crisis.

In announcing his coronavirus task force’s 5 p.m. news conference, the president singled out Mr. Biden, the current front-runner in the Democratic presidential primary.

“The USA was never set up for this, just look at the catastrophe of the H1N1 Swine Flu (Biden in charge, 17,000 people lost, very late response time),” Mr. Trump tweeted. In fact, under the Obama administration, diagnostic tests for the 2009 swine flu outbreak were approved and shipped less than two weeks after the H1N1 virus was identified and a day before the first death in the United States.

And even as a top health official warned that Americans should prepare to “hunker down significantly more than we as a country are doing,” Mr. Trump was focused on Mr. Schumer, the Senate minority leader. Resuscitating a story that made the rounds in conservative news media weeks earlier, the president accused Mr. Schumer of threatening his two appointees to the Supreme Court.

“Can’t believe they are not going after Schumer for the threats he made to our cherished United States Supreme Court, and our two great Justices,” Mr. Trump tweeted. “If a Republican did that, there would be an endless price to pay. Pathetic!” He was referring to comments Mr. Schumer made this month, when he said Justice Brett Kavanaugh and Justice Neil M. Gorsuch had “released the whirlwind, and you will pay the price! You won’t know what hit you if you go forward with these awful decisions.”