Washington (CNN) -- The chairman of the Republican Party and a leading GOP senator called on Senate Majority Leader Harry Reid to give up his post Sunday, following the publication of remarks he made about President Obama's race in 2008.

A new book quotes Reid, D-Nevada, as saying privately in 2008 that Obama could be successful as a black candidate in part because of his "light-skinned" appearance and speaking patterns "with no Negro dialect, unless he wanted to have one."

The remarks were "embarrassing and racially insensitive," said Sen. John Cornyn, R-Texas, head of the GOP's Senate campaign arm, in a statement to CNN.

GOP Chairman Michael Steele, on NBC's "Meet the Press," said: "Racism and racist conversations have no place today in America."

Steele also was on the defensive for a remark he made last week that members of both parties have called a racial slur. In an interview with Fox News, Steele used the phrase "honest injun."

The Congressional Black Caucus has accepted Reid's apology and is dismissing calls for him to step down as majority leader.

Rep. Barbara Lee, chairwoman of the caucus, issued the following statement: "I have had an opportunity to speak with Senator Reid and he apologized for his unfortunate remarks concerning the president, and he understands the gravity of such remarks. There are too many issues like the economy, job creation and energy for these regrettable comments to distract us from the work that must be done on behalf of the American people."

Democrats also rejected the calls for Reid's dismissal. Gov. Tim Kaine of Virginia, chairman of the Democratic Party, said "the case is closed" following Reid's round of apologies.

Douglas Wilder of Virginia, who 20 years ago became the nation's first elected African-American governor, also rejected calls for Reid's ouster.

"I think that what Reid was giving was a personal opinion, which wasn't affecting the laws or the operation of the dispensation of justice in our country," Wilder told CNN's "State of the Union."

But he said he believes the incident "illustrates the need for more open discussion about race."

Reid's office made clear he has no plans to step down.

Democrats rejected the calls for Reid's dismissal, and Reid's office made clear he has no plans to step down.

"Sen. Reid will stay in his position as majority leader and will run for re-election," his spokesman said.

"As the leader in the fight to pass the Voting Rights Act and legislation banning hate crimes, Sen. Reid has a long record of addressing issues that are important to the African-American community. His Republican critics who are looking to politicize the issue can't say the same."

Reid's controversial quote is in the book "Game Change," due in stores Monday. The authors write that "Reid was convinced, in fact, that Obama's race would help him more than hurt him in a bid for the Democratic nomination."

In a statement to CNN, Reid said, "I deeply regret using such a poor choice of words."

"I sincerely apologize for offending any and all Americans, especially African-Americans for my improper comments. I was a proud and enthusiastic supporter of Barack Obama during the campaign and have worked as hard as I can to advance President Obama's legislative agenda," Reid said.

In his defense, he pointed to his efforts to integrate the Las Vegas strip and the gaming industry, among other legislation favored by African-American voters.

"I have worked hard to advance issues important to the African-American community," he said.

And the senate leader called Obama on Saturday afternoon to apologize for the remarks. In a statement issued after the call, Obama said, "As far as I am concerned, the book is closed."

"Harry Reid called me today and apologized for an unfortunate comment reported today," the president said. "I accepted Harry's apology without question because I've known him for years, I've seen the passionate leadership he's shown on issues of social justice and I know what's in his heart."

An aide to the senator told CNN that Reid also offered apologies to several prominent African-American political figures, including House Democrats Jim Clyburn of South Carolina and Barbara Lee of California; the Rev. Al Sharpton; CNN political contributor and Democratic strategist Donna Brazile; NAACP chairman Julian Bond; and the head of the Leadership Conference on Civil Rights, Wade Henderson.

Steele, the GOP's first African-American chairman, was asked about the remarks on both "Meet the Press" and "Fox News Sunday." He told NBC on Sunday he believes Reid is out of touch with "how African-Americans generally feel" about sensitive issues.

Steele was asked by NBC whether he believes the situation is similar to one involving former Sen. Trent Lott, who lost his post as Senate majority leader in 2002 after saying that the nation would have been better off if one-time segregationist candidate Strom Thurmond had been elected president.

"Oh, yeah. There is a big double standard here," Steele said on NBC.

Steele added: "When Democrats get caught saying racist things, you know, an apology is enough."

Steele said that if a Republican senator had made the same remark Reid did, Steele himself and the Democratic Party "would be screaming for his head very much as they were with Trent Lott."

Cornyn, in his statement, also accused Democrats of following a "double standard," and noted that they had pushed Lott to step down.

"As we await his explanation, Sen. Reid should do the right thing, follow the example that he himself set in 2002, and step down as majority leader," Cornyn said.

Kaine shot back against those arguments.

"Anybody looking at Trent Lott's statements praising somebody who had been a pro-segregation candidate for president will see that there is no comparison between those comments and those of Sen. Reid," Kaine told NBC.

The comments "were in the context of praising the senator and acknowledging that the senator could be a great president, but they were still insensitive," Kaine said.

Asked whether Reid should resign, he said, "Absolutely not. ... We're moving on."

Nevada state Senate Majority Leader Steven Horsford, who is African-American, affirmed his support for Reid in a statement Saturday.

"While I am disappointed in Sen. Reid's comment and choice of words, I accept his apology," said Horsford, a Democrat. "I have known Sen. Reid for many years and he has consistently been supportive of advancing the interests of the African-American community as he has for all Nevadans and all Americans."

Steele, meanwhile, was asked about his remark in a Fox News interview last week that the GOP platform "is one of the best political documents that's been written in the last 25 years, 'honest injun' on that."

"Fox News Sunday" host Chris Wallace noted that lawmakers from both parties have called that a racial slur.

"Well, if it is, I apologize for it. It's not an intent to be a racial slur. I wasn't intending to say a racial slur at all," Steele said.

CNN's Dana Bash, Mark Preston and Rebecca Sinderbrand contributed to this report.