Speaking with Joy-Ann Reid of MSNBC on Friday night, Mr. Steele, a former lieutenant governor of Maryland and the first black person elected to statewide office there, said he did not accept an apology from Mr. Walters.
“He said at one point, ‘I apologize,’ Mr. Steele said, adding that he told Mr. Walters, “That’s not enough.”
Mr. Walters apologized on Twitter on Friday night. “I spoke with Mike Steele and apologized because the words I used do not capture my heart,” he wrote.
He added: “Sometimes when you speak, the words that come out do not reflect what’s intended. Many of us were critics of how Chairman Steele performed at the R.N.C. He is a good man, and he did his best.”
In a phone interview on Saturday morning on Ms. Reid’s show, Mr. Steele was asked if the Republican Party had a problem with racism.
“Yes, they do. And I think we need to be honest and acknowledge it,” Mr. Steele said. “The fact that people sit here now and say this has nothing to do with race — yeah, it does, when you stand on a podium and blatantly speak to race the way Ian did.”
Mr. Walters did not respond to a request for comment on Saturday.
Michael Ahrens, a spokesman for the Republican National Committee, said in a statement, “We reject the offensive comments made last night.”
Black people made up 2 percent of registered Republican voters in 2016, according to the Pew Research Center, the same percentage as in 1992. President Trump received 8 percent of the black vote in 2016.
Mr. Steele, speaking Saturday on his show on SiriusXM radio with Matt Schlapp, the chairman of the American Conservative Union, said comments like Mr. Miller’s “undermine” efforts to “expand this party and its reach into communities of color across the country.”
Mr. Schlapp apologized to Mr. Steele and said Mr. Walters’s comments were “unfortunate words” that might have been based on Mr. Steele’s criticism of the “Trump phenomena.”
Mr. Steele has been critical of Mr. Trump’s racially tinged comments about Haiti and some African nations. Mr. Steele also said that Mr. Trump’s suggestion that armed teachers would help prevent school shootings was “desperate and delusional.”
On Saturday, Mr. Schlapp urged Mr. Steele “to have some grace,” which angered the host.
“Wait a minute,” Mr. Steele said. “What the hell do I have to be graceful for?”
He added: “It is stupid to sit there and say we elected a black man chairman of the party and that was a mistake. Do you know how that sounds to the black community? Do you know how that sounds to Americans?”