In a tweet on Thursday, Trump said "publicity seeking" Graham incorrectly stated that the president said "there is moral equivalency between the KKK, neo-Nazis & white supremacists. ... and people like Ms. Heyer."
Heather Heyer was killed after she was struck by a car driven into a crowd of people who were protesting a white nationalist rally in Virginia.
In a statement on Wednesday, Graham said Trump "took a step backward by again suggesting there is moral equivalency between the white supremacist neo-Nazis and KKK members who attended the Charlottesville rally and people like Ms. Heyer. I, along with many others, do not endorse this moral equivalency."
Trump on Monday explicitly condemned white supremacists and "racist" violence, two days after he condemned hatred and violence "on many sides" without specifically denouncing white supremacist groups.
At a heated news conference on Tuesday, Trump doubled down on his initial response, blaming "both sides" for the violence.
In a separate tweet, Trump took a shot at the "fake news" media, saying mainstream media is distorting "what I say about hate, bigotry, etc. Shame!"