BRIDGEWATER, N.J. — White House officials, under siege over President Trump’s reluctance to condemn white supremacists for the weekend’s bloody rallies in Charlottesville, Va., tried to clarify his comments on Sunday, as critics in both parties intensified demands that he adopt a stronger, more unifying message.

A statement on Sunday — issued more than 36 hours after the protests began — condemned “white supremacists” for the violence that led to one death. It came in an email sent to reporters in the president’s traveling press pool, and was attributed to an unnamed representative.

It was not attributed directly to Mr. Trump, who often uses Twitter to communicate directly on controversial topics. It also did not single out “white supremacists” alone but instead included criticism of “all extremist groups.”

The email was sent “in response” to questions about Mr. Trump’s remarks, in which he blamed the unrest “on many sides” while speaking on Saturday before an event for military veterans at his golf resort in Bedminster, N.J., where the president is on vacation.

“The president said very strongly in his statement yesterday that he condemns all forms of violence, bigotry and hatred,” the statement said. “Of course that includes white supremacists, K.K.K. neo-Nazi and all extremist groups. He called for national unity and bringing all Americans together.”

Mr. Trump’s high-volume outbursts on issues petty and profound have become a defining feature of his presidency. But his quiescence on the violence in Charlottesville has had, in many ways, a more profound and unsettling effect.

The president’s reluctance to speak out with force and moral indignation against the white nationalists who incited the most serious racial episode of his presidency elicited deep feelings of disappointment spanning the ideological spectrum, and a spreading sense that he had squandered a critical opportunity to empathize, unite and move beyond the acrimony that has engulfed the White House and country.

“I think what you saw here was a real moment in our nation for our leaders to deal with this moral issue as one country, as people all over the world watched,” Gov. Terry McAuliffe of Virginia said, speaking on a cellphone outside the home of one of the two state troopers killed in a helicopter crash monitoring the melee on Saturday.

Mr. Trump’s “words were not — not — what this nation needs,” Mr. McAuliffe, a Democrat, said, his voice breaking with emotion. “He needs to call out the white supremacists, he needs to call out the neo-Nazis to say these people should not be in our country. I do think it’s the president’s responsibility to take leadership on this. It’s what any American would do. Now is the time to step up.”

Gov. John Kasich of Ohio, a Republican and a frequent critic of Mr. Trump, cautioned against reading too much into the president’s initial response but called for the White House to use the episode as an opportunity to convene “a national discussion” on race, prejudice and community policing.

“There are a lot of people who are just not comfortable with the issue; perhaps they are afraid it would aggravate their base,” Mr. Kasich said, adding, “I think a president can always provide some leadership on a subject like this.”

The criticism of Mr. Trump intensified on Sunday, with lawmakers from both parties calling on him to explicitly condemn the role of white racists and agitators affiliated with the fringe movement known as the alt-right, some of whom brandished pro-Trump banners and campaign placards during violent protests over the removal of a Robert E. Lee statue from a Charlottesville park.

Mr. Trump will continue to receive regular updates from his team, according to the official to whom the statement was attributed, and Thomas P. Bossert, the White House Homeland Security adviser, was in Bedminster monitoring the situation.

As the White House shifted its message, the Justice Department opened a hate crimes inquiry into the violence, which included the death of a 32-year-old woman.

James Alex Fields Jr. of Ohio was charged with second-degree murder, accused of running down her and others in a car. Nineteen other people were injured in the episode, which Mr. McAuliffe called “murder, plain and simple.”

Attorney General Jeff Sessions was pressed during his confirmation hearings early this year about how he might handle such a case, and many on Sunday said they saw the Charlottesville investigation as a test for him.

In a statement late Saturday, Mr. Sessions went further than the president had in his remarks, condemning not just the violence and deaths in Charlottesville but adding that “when such actions arise from racial bigotry and hatred, they betray our core values and cannot be tolerated.”

It was unclear on Sunday what the specific scope of the investigation was. A spokesman for the department declined to comment beyond a statement released late Saturday that had cited the cooperation of the F.B.I. field office in Richmond, Va.; the United States Attorney’s Office for the Western District of Virginia; and the Justice Department’s civil rights division.

“It’s a test for Attorney General Sessions in that he has put such a premium on cracking down on violence in cities and violent offenders, and obviously during his confirmation there were issues raised about equal justice under law,” said John P. Fishwick, a former United States attorney for the Western District of Virginia during the Obama administration.

Administration officials largely appeared to avoid branding the car rampage as domestic terrorism, but some experts said it certainly could meet the legal standard.

“You can be a domestic terrorist if you’re trying to intimidate the population, or if you’re trying to affect the conduct of the government by mass destruction,” Mr. Fishwick said. “Nineteen people wounded — a number of them critically — and one dead. I think that’s enough.”

Mr. Trump consulted a broad range of advisers before speaking on Saturday, most of whom told him to sharply criticize the white nationalist protesters.

At the center of the discussion was Mr. Bossert, who laid out the situation on the ground, including a description of provocations by both protesters and counterprotesters, according to a White House official.

Two hard-edge economic populists — Stephen K. Bannon, the president’s chief strategist, and Stephen Miller, a senior adviser — spoke with Mr. Trump repeatedly on Saturday, the person said, although it was not clear if Mr. Bannon had offered him advice on his comments.

Mr. Trump listened attentively, according to another person familiar with the discussions, but repeatedly steered the conversation to the breakdown of “law and order,” and the responsibility of local officials to stem the violence.