For the Secret Service, a New Question: Who Will Protect Them From Trump?

The drive-by visit to Mr. Trump’s supporters drove home that point, Ms. Napolitano said.

“There was nothing about his little motorcade yesterday that was required or wise or necessary,” she said on Monday. “It was a superfluous act.”

Secret Service agents have always needed to make last-minute adjustments based on the political whims of presidents, said Michael Beschloss, a presidential historian. And the relationship between presidents and their protection details has usually allowed for a limited amount of debate. But in the end, presidents have an appetite for making public appearances before their supporters, allies and in some cases, adversaries. And the president has final say.

“These are people that have volunteered to give their lives for theirs, and almost every president I can recall uses that privilege carefully and with great respect,” Mr. Beschloss said.

Mr. Trump responded to the criticism on Monday by blaming the news media.

“I got into a secure vehicle to say thank you to the many fans and supporters who were standing outside of the hospital for many hours, and even days, to pay their respect to their President,” he posted on Twitter. “If I didn’t do it, Media would say RUDE!!!”

Rand Beers, a former acting secretary of the Department of Homeland Security, acknowledged that presidents were obliged to convey “to the nation that things are on track.”

“But,” he added, “you’re still putting agents at risk if you’re not taking precautions.”

The day after Mr. Trump’s ride, Representative Bennie Thompson, Democrat of Mississippi and the chairman of the House Homeland Security Committee, demanded a briefing from the Department of Homeland Security to learn more about the safeguards in place for the agency’s employees and specifically the protection detail.

“The height of reckless disregard for others was the president’s joy ride yesterday, where Secret Service agents were required to drive him around in a hermetically sealed vehicle,” Mr. Thompson said. “Exposing Secret Service personnel to the virus does not just put them at risk, it puts their families and the public at risk.”

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