The rally erupted into deadly violence as a car plowed into a crowd while demonstrators and counter-protesters clashed.
US authorities on Sunday were investigating the outbreak of violence in Virginia following a white nationalist rally that killed one person and injured 19.
Virginia police have not yet provided a motive for a man plowing a car into a crowd of people objecting to the white nationalists, but US attorneys and the Federal Bureau of Investigation have opened a civil rights investigation into the crash, an FBI field office said.
TRT World's Tetiana Anderson has more.
Four people have been arrested, including James Fields, a 20-year-old white man from Ohio who is being held in jail on suspicion of crashing the car. The vehicle killed a 32-year-old woman and injured 19 people, five of them critically.
Federal authorities were also looking into a helicopter crash on Saturday that killed two Virginia state policemen aiding efforts to quell the clashes.
The violence in the Southern college town of Charlottesville on Saturday was widely condemned, with many politicians and activists on both the left and right also criticising Trump, a Republican, for waiting too long to address it and when he did so, failing to explicitly condemn the white-supremacist marchers who ignited the melee.
A domestic challenge for Trump
Trump said on Saturday that there was more than one side to the Charlottesville incidents.
"We condemn, in the strongest possible terms, this egregious display of hatred, bigotry and violence on many sides," he told reporters at his New Jersey golf course, without specifically mentioning or faulting the role of white nationalists.
On Sunday, however, the White House said in a statement that Trump's remarks on Saturday condemned all forms of violence and bigotry, including neo-Nazi groups and the Ku Klux Klan.
"The president said very strongly in his statement yesterday that he condemns all forms of violence, bigotry, and hatred, and of course that includes white supremacists, KKK, neo-Nazi, and all extremist groups," the White House said. "He called for national unity and bringing all Americans together."
On Sunday morning, before the White House statement, Ivanka Trump, the president's daughter and White House adviser, appealed on Twitter for Americans to "be one country UNITED. #Charlottesville."
1:2 There should be no place in society for racism, white supremacy and neo-nazis.— Ivanka Trump (@IvankaTrump) August 13, 2017
An organiser of Saturday's "Unite the Right" rally, which was staged to protest the planned removal of Confederate war hero Robert E. Lee's statue from a park, said supporters of the event would not back down.
"Absolutely we are going to have further demonstrations in Charlottesville because our constitutional rights are being denied," said Jason Kessler, whom civil rights groups identified as a white nationalist blogger. He did not specify when.
Long history of violence
The Charlottesville violence is the latest clash between far-rightists, some of whom have claimed allegiance to Trump, and the president's opponents. At his January inauguration, black-clad anti-Trump protesters in Washington smashed windows, torched cars and clashed with police, leading to more than 200 arrests.
The rally stemmed from a long debate in the US South over the Confederate battle flag and other symbols of the rebel side in the Civil War, which was fought over slavery.
About two dozen people were arrested in Charlottesville in July when the Ku Klux Klan rallied against the plan to remove the Lee statue. Torch-wielding white nationalists also demonstrated in May against the removal.