At a news conference earlier in the day, Trump defended the protest that led to the violence and contended that some of the individuals carrying torches at the rally did not have bad intentions. He blamed all sides for the violence in Charlottesville, Virginia.
"You had many people in that group other than neo-Nazis and white nationalists and the press has treated them absolutely unfairly," Trump said.
He added, "I think there's blame on both sides and I have no doubt about it and you don't have any doubt about it either."
McCain, who was the GOP's 2008 presidential nominee, said on Twitter that, as president of the United States, Trump should make clear that there was "no moral equivalency between racists & Americans standing up to defy hate& bigotry."
Meanwhile, the 2012 Republican nominee Mitt Romney also condemned Trump's response at the earlier press conference. In a tweet, Romney said, "No, not the same. One side is racist, bigoted, Nazi. The other opposes racism and bigotry. Morally different universes."
Scuffles broke out between participants at the white nationalist rally and counterprotesters over the weekend. A suspected white nationalist allegedly later rammed a car into a group of counterprotesters, killing one and injuring 19 people.
Following the incident, Trump had initially condemned violence "on many sides." The move drew bipartisan criticism, and the White House then attempted to limit the damage on Monday with Trump making a statement condemning neo-Nazis, white supremacists and KKK members.
— CNBC's Jacob Pramuk contributed to this report.