Over all, Americans are skeptical of Mr. Trump’s ability to confront the crisis. Last month, before the virus had spread throughout the United States, 62 percent of voters nationwide said they were not very confident in the federal government’s ability to deal with it effectively, according to a Fox News poll. Democratic voters were more than twice as likely as Republicans to say that they had no confidence at all.

For weeks, Mr. Trump oscillated between reassurances that he had the situation under control and suggestions that the whole issue was vastly overblown. Democrats, including Speaker Nancy Pelosi and the party’s leading presidential candidate, former Vice President Joseph R. Biden Jr., have called the situation grave and pushed for an aggressive response.

Mr. Biden and his leading rival, Senator Bernie Sanders of Vermont, canceled in-person rallies this week and announced online campaign events in their stead. On Thursday, both candidates strongly criticized Mr. Trump’s address to the nation.

“Public fears are being compounded by pervasive lack of trust in this president,” Mr. Biden said. He offered his own plan to combat the crisis while condemning what he called Mr. Trump’s “colossal” failure to guarantee citizens access to coronavirus testing.

For Mr. Trump, the perception that he is a strong leader has been central to his appeal since the beginning of the 2016 presidential campaign. Gallup regularly asks Americans about their feelings about the president on a range of personal qualities. Out of the seven qualities Gallup tests, only one applies to Mr. Trump in the eyes of a majority of Americans: being a strong and decisive leader. When Mr. Biden attacks him over the coronavirus, he is taking a cut at Americans’ view of Mr. Trump as a forceful leader.

But the virus may have an effect on Mr. Trump’s public approval in ways that ripple far beyond his direct response to the crisis — particularly as the economy absorbs the severe impact of the virus.

Before the outbreak, Americans were more likely to describe themselves as satisfied with the way things were going in the country than at any point in the past 15 years, according to Gallup polling. Mr. Trump’s approval ratings have been in the mid- to high 40s in recent months — among the highest numbers of his presidency — lifted by the thriving economy. Americans generally tend to approve of his handling of the economy more than of his overall performance.