"To get into an argument about who has done the most probably is not productive, but it is good to acknowledge the things that have been done," Carson said on Sunday.
Pressed by ABC News Chief Anchor George Stephanopoulos on Trump's claim, Carson said it's an "important thing for us to acknowledge what has happened in the past," while also touting Trump's record on criminal justice reform and funding for historically black colleges and universities.
"We should be willing to look at what we've done collectively together to make progress," he said.
The only African American cabinet secretary in the Trump administration, Carson said Americans need to stop "putting everything in the arena of combat" and "find a way to work together" following George Floyd's murder in police custody, and the ongoing protests over racial inequality and police brutality.
"We obviously need to acknowledge that there's a reason that the protests are going on," he said. "But it also means we need to open the discussion, we need to listen to the police as well as the protesters."
Questioned by Stephanopoulos on Trump's repeated calls for law enforcement to "dominate the streets," Carson suggested Trump was saying "we cannot submit to anarchy."
"I think we would all agree with that," he said. "We might not all express it the same way but we would all agree with it."
Carson praised the president's decision to move his planned Tulsa, Oklahoma, rally from Juneteenth. Tulsa was also the site of a bloody massacre of African-Americans by a white mob in 1921 -- one of the worst episodes of violence against black Americans in the country's history.
"It's probably good to have moved it," said Carson, who noted that he spoke to Trump about moving the rally and was "pleasantly surprised" by how much he knew about Juneteenth.
But he dismissed questions about new plans for Trump to accept the Republican Party's nomination for president in Jacksonville, Florida, on the 60th anniversary of "Ax Handle Saturday," when a white mob and members of the Klu Klux Klan chased down and beat African Americans, following a lunch counter sit-in protesting segregation.
"Some of our prestigious universities have a relationship with the slave trade. Should we go and rename those universities?" he said. "It really gets to a point of being ridiculous after a while, and, you know, we're going to have to grow up as a society."
The retired neurosurgeon also supported Trump's plans to hold the rally in Oklahoma despite the increasing number of COVID-19 cases in the city. The director of the Tulsa Health Department has said he "wishes" the rally would be postponed.
"I think if it's done in conjunction with the public health experts, which it is being done in conjunction with them, that is quite acceptable," Carson said.
"It is very important that we utilize what we have learned about the disease so that we can live with it, rather than allow it to dominate us," he said.
The Trump campaign has told ABC News that safety precautions will be in place at the rally, but has yet to provide any details.
Health experts have warned that the virus can spread easily in crowded, indoor venues. The Trump campaign added a liability waiver to the sign-up page for the rally on its website, which would absolve the campaign and venue if any attendees contract COVID-19.