Trump on Thursday called Graham a "publicity seeking" lawmaker and tweeted that Graham's contention that the president had said there was "moral equivalency between the KKK, neo-Nazis & white supremacists and people like Ms. Heyer" was a "disgusting lie".
Heather Heyer was the counter-protester killed in a vehicle attack Saturday on the sidelines of the white nationalist rally in Virginia.
Trump has been on the defensive in the past several days following his comments Saturday that blamed "both sides" for the chaos that erupted in the streets of the Virginia town.
Trump fired back Thursday, claiming Graham's statements were lies.
Trump came out Monday and specifically condemned white supremacist groups like neo-Nazis and the Ku Klux Klan after only criticizing "many sides" on Saturday.
Soundcloud raises new funding, names former Vimeo exec as CEO
Trainor said this open artistic experimentation gives the platform a unique identity in the digital music space. Co-founder and Chief Technology Officer Eric Wahlross will remain at the company as chief product officer.
Trump had criticized Graham earlier via Twitter, issuing a pair of tweets early in the morning August 17.
Chairman of the Joint Chiefs of Staff Joseph Dunford has strongly endorsed the statements by the leaders of the four major US military services, who spoke out against racism and extremism after last weekend's violence in Charlottesville, Virginia.
Graham, a frequent critic of the President who unsuccessfully ran against him in the 2016 presidential primaries, warned Wednesday that Trump's rhetoric is "dividing Americans". A few hours later, Graham released a statement in response to the president's tweets.
Trump has faced bipartisan criticism for his back-and-forth comments on Charlottesville, both decrying white supremacist groups and then on Tuesday backtracking by placing blame on liberal groups in addition to white nationalists.
President Donald Trump speaks to the media in the lobby of Trump Tower, Tuesday, Aug. 15, 2017 in NY. "They totally misrepresent what I say about hate, bigotry etc. Shame!" Trump said in an interview this week he was considering pardoning Arpaio, who was one of the speakers at last year's Republican National Convention where the now-president accepted the party's nomination for the November election. "I think it is important for all those in positions of responsibility to condemn far-right views wherever we hear them". Lindsey Graham on notice for his criticism of: "The people of SC will remember!"