"I don't know anything about what you're even talking about with white supremacy or white supremacists," Trump said. "So, I don't know. I don't know. Did he endorse me? Or what's going on? Because, you know, I know nothing about David Duke. I know nothing about white supremacists. And so you're asking me a question that I'm supposed to be talking about people that I know nothing about."
What Donald Trump is doing on the campaign trail
Trump told Tapper to send him a list of groups and he would then provide assessments of each, saying that he doesn't want to condemn a group about which he knows nothing.
"Certainly, I would disavow if I thought there was something wrong," Trump said.
Tapper then offered one group: "The Ku Klux Klan?" He further clarified: "I mean, I'm just talking about David Duke and the Ku Klux Klan here."
"I don't know any — honestly, I don't know David Duke," Trump said. "I don't believe I have ever met him. I'm pretty sure I didn't meet him. And I just don't know anything about him."
Trump's responses quickly prompted criticism from his Republican rivals and others.
"He knows exactly who David Duke is," Sen. Marco Rubio said as he rallied with thousands at Patrick Henry College in Purcellville, Va. "He was asked this morning two times — will you repudiate and condemn the Ku Klux Klan, and he refused that, as well. We cannot be a party who nominates someone who refuses to condemn white supremacists and the Ku Klux Klan."
Rubio said Trump would be "unelectable" as a GOP nominee if he failed to repudiate Duke. He also claimed that Trump knew of Duke, noting that the former declined to run as a Reform Party candidate several years ago because Duke was seeking the party's nomination.
Sen. Ted Cruz (R-Tex.) called Trump's comments "really sad" and wrote in a tweet that Trump is "better than this."
"We should all agree, racism is wrong, KKK is abhorrent," Cruz wrote.
Speaking to reporters in Oklahoma City, Cruz said it is "unfortunate" that Trump declined to denounce Duke, "particularly when layered on the fact that white nationalists were doing robocalls urging people to vote for Donald Trump and telling people, 'don’t vote for the Cubans in this race,'" Cruz said.
The Texas Republican added: "In my view, racism and bigotry has no place in politics. And we should all be united in saying that the Klan is reprehensible and has no place in politics ..... That should be common ground among Republicans, among Democrats, among everyone, that bigotry that antisemitism, that racism has no role in politics."
Ohio Gov. John Kasich mentioned Trump's comments as he started a town hall in Springfield, Mass.
"Apparently he refused to disassociate himself from white supremacists," Kasich said. "Every day it's another thing. That's just horrific, right? We don't have any place for white supremacists in the United States of America. It doesn't make any sense. He really needs to make his position clear, and he ought to do it quickly."
Early Sunday afternoon, Trump tweeted a video of himself at a news conference on Friday saying: "I didn't even know he endorsed me. David Duke endorsed me? Okay, all right. I disavow. Okay?" Trump wrote in his tweet: "I disavow."
Ed O'Keefe, Katie Zezima and David Weigel contributed to this report.