The driver, James Alex Fields Jr., a 20-year-old from OH, was charged with second-degree murder.
Attorney General Jeff Sessions said the violence, "strikes at the heart of American law and justice".
Ted Cruz is calling on the Justice Department to investigate the driver of a auto that plowed through a crowd of protesters in Charlottesville, Va. on Saturday, an act which the Republican is calling domestic terrorism. "When such actions arise from racial bigotry and hatred, they betray our core values and can not be tolerated".
Late Saturday night, the FBI in Richmond, Va. said it had opened an investigation.
Fields, who is from Maumee, Ohio, allegedly drove his 2010 Dodge Challenger through a group of counter-protesters gathered in Charlottesville to demonstrate against thousands of white nationalists and neo-Nazis who were holding a rally to protest the removal of a Robert E. Lee statue in a local park.
Spencer said he recommended that people should disperse after the state of emergency was declared. Officials declined to comment further.
Sessions promised a full federal investigation of the clashes and ending his statement Saturday saying, "Justice will prevail". Now, Sen. Ted Cruz (R-TX) is demanding the Department of Justice investigate the vehicle crash as an act of domestic terrorism.
An Obama-era Justice Department official also called for federal investigators to look into Saturday's violence.
Fields has been charged with second-degree murder in the case.
Three other people were arrested in connection with the violence, CNN reported, though it was not clear whether they had a connection to the deadly vehicle crash.
Troy Dunigan, 21, of Chattanooga, Tennessee was charged with disorderly conduct; 21-year-old Jacob L. Smith, of Louisa, Virginia, was charged with misdemeanor assault and battery; and 44-year-old James M. O'Brien of Gainesville, Florida was charged with carrying a concealed handgun.