Ohio Gov. Mike DeWine said late Monday that the state department of health will order polls to be closed on Tuesday as a “health emergency,” hours after a judge declined to delay the state’s March 17 presidential primary at the behest of Mr. DeWine.

Mr. DeWine, a Republican, said that to conduct an election Tuesday would force poll workers and voters to place themselves “at an unacceptable health risk of contracting coronavirus.”

He tweeted that Dr. Amy Acton, who heads the Ohio Department of Health, “will order the polls closed as a health emergency.”

“While the polls will be closed tomorrow, Secretary of State @FrankLaRose will seek a remedy through the courts to extend voting options so that every voter who wants to vote will be granted that opportunity,” Mr. DeWine said.

Dr. Acton signed a three-page order saying that all polling locations in the state would be closed “to avoid an imminent threat with a high probability of widespread exposure to COVID-19 with a significant risk of substantial harm to a large number of people in the general population.”

The order said she had authority under Ohio code to “make special orders…for preventing the spread of contagious or infectious diseases.”

The action came hours after Franklin County Judge Richard Frye denied a motion to push in-person voting to June 2 amid concerns over the coronavirus outbreak, saying such a move could set a terrible precedent.

After the ruling, Mr. DeWine and Secretary of State Frank LaRose issued a joint statement saying that given the circumstances, “it simply isn’t possible to hold an election tomorrow that will be considered legitimate by Ohioans.”

“They mustn’t be forced to choose between their health and exercising their constitutional rights,” the statement said.

Earlier in the day, Judge Frye had said that decision wasn’t his to make, saying he was “very reluctant” to “say well, we’ll have a judge in Columbus rewrite the election code, reset the election for some arbitrary date in the future and upset the apple cart in a terrible precedent.”

Mr. DeWine spoke as voters in Ohio, Arizona, Florida, and Illinois were preparing to head to the polls to cast their ballots in a race that has been upended by concerns over the coronavirus that is changing the daily lives of millions of Americans.

President Trump, who said on Monday that Americans should avoid gatherings of more than 10 people for the next 15 days as part of the coronavirus response, said postponing elections is a state matter but that he generally isn’t a fan.

“I think postponing elections is not a very good thing,” Mr. Trump said. “They have lots of room in a lot of the electoral places. … I think postponing is unnecessary.”

Florida Gov. Ron DeSantis said Monday that elections can be done in a way that protects people.

“We’ve limited access to the assisted living facilities in some of the areas where we may have vulnerable folks,” Mr. DeSantis said. “But at the same time, when you go and cancel that, the signal that that sends is somehow we’re paralyzed. And I don’t think that’s the case. I think we’re taking prudent steps.”

Arizona Secretary of State Katie Hobbs said Monday that officials understand voters might be apprehensive, but the election would proceed as planned.

“This decision was not made lightly, and what it all comes down to is — that we have no guarantee that there will be a safer time to hold this election in the near future,” Ms. Hobbs said. “The longer we wait, the more difficult and dangerous this will become.”

A spokesman for the Illinois State Board of Elections likewise said the state primary was proceeding as planned, saying the elections team has been working with public health officials and has made every effort to keep voters up to date on polling place location changes.

Dan Kovats, executive director of the Illinois Democratic County Chairs’ Association, said Illinois has done a good job encouraging early voting and voting by mail over the last few cycles.

“I believe we will have everything in place to ensure a legitimate primary election,” Mr. Kovats said.

He said his group, in concert with the state Democratic Party, launched a large-scale texting program Monday morning to inform voters of changes in their polling locations and that he thought more than 300 precincts have had such changes.

Officials in all four states already had been struggling to convince poll workers to show up and grappling with polling locations getting cut back or changed amid the escalating outbreak.

Before Mr. Trump’s announcement, the Centers for Disease Control and Prevention had issued guidance late Sunday recommending that the public avoid gatherings of 50 or more people for the next eight weeks.

In Florida, some local officials had already said they were short of where they wanted to be in terms of attracting enough workers to man the polls Tuesday in light of COVID-19.

Rep. Debbie Mucarsel-Powell, Florida Democrat, said Monday the state should have followed Ohio’s lead and pushed back its Tuesday election as well.

“We need to protect the rights of all our citizens to safely have their voices heard,” Ms. Mucarsel-Powell tweeted, as she also expressed concern about reports of a lack of mail ballots in South Dade County.

In Arizona, Maricopa County was planning to close dozens of polling locations because of COVID-19.

“The closure of many polling places due to COVID-19 means it is uncertain how many voters who planned to vote on Tuesday will actually show up,” said Patrick Murray, director of the Monmouth University Polling Institute.

Kentucky Gov. Andy Beshear said Monday his state will push its primary from May 19 to June 23, become the latest state leader to announce such a change.

Louisiana and Georgia have already postponed their primaries that had been scheduled to take place in the coming weeks, and the Democratic Party of Puerto Rico said it was going to push for a delay in its March 29 primary.

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