But Pence declined to categorize Trump backer -- and white nationalist -- David Duke as "deplorable."
"I'm not in the name-calling business," Pence told CNN's Wolf Blitzer, while at the same time repeating Trump's disavowal of Duke's support.
"We don't want his support and we don't want the support of the people who think like him," he said.
Pence repeated his comments during a Tuesday news conference on Capitol Hill.
"I have no idea why this man keeps coming up," the Indiana governor said. "Donald Trump and I have denounced David Duke repeatedly. We have said that we do not want his support and we don't want the support of people who think like him."
Clinton jumped on Pence's response, tweeting: "If you won't say the KKK is deplorable, you have no business running the country."
Her running mate, Tim Kaine, also hit Pence over the remark, telling supporters at a campaign event in Ann Arbor, Michigan, on Tuesday, "If you cannot call out bigotry, if you cannot call out racism, xenophobia, anti -- if you can't call it out and you stand back and you're silent about it, you're enabling it to grow, you're enabling it to become more powerful."
Clinton later said she regretted using the word "half" to describe the percentage of Trump supporters she feels are racist, bigoted or xenophobic, but said she did not regret calling out Trump's more unseemly supporters.
"Hillary Clinton did not apologize for insulting millions of people," Pence told Blitzer on "The Situation Room," calling her speech at a fundraiser "intentional" and "forthright."
"Did she walk it back? She said she regretted saying half. So what was it -- 40%?" Pence said.
Pence also said that both Clinton and Trump should release detailed medical information on their physical health, arguing that the disclosures are in the public interest.
"People are vying for the highest office in the land," the Indiana governor said. "People have a right to know."
Clinton is taking a day off the campaign trail, staying at her home in Chappaqua after stumbling at a 9/11 commemoration event on Sunday. Aides later said she was diagnosed Friday with pneumonia.
Pence is in Washington ahead of meetings with Capitol Hill Republicans on Tuesday.
Asked if top Trump adviser Rudy Giuliani had gone too far in questioning Clinton's health prior to Sunday's incident, Pence demurred.
"We wish Hillary Clinton well. We're praying for a swift recovery. We want to see her back on the campaign trail," he said. "We'll leave it at that."
Pence also defended his running mate's reluctance to release his tax returns, even though Pence himself just released reams of financial data about his own income and tax rates. Trump has said that he will not share his until a "routine audit" is completed -- and Pence said the Republican nominee has every intention of doing so.
Pressed as to why Trump would not release topline information about previous returns now -- which would not interfere with the audit process -- Pence said Trump would release his returns "in totality" and "not parse them out piece by piece."
Pence added that Trump was not violating any laws by withholding the data, though he acknowledged "there's a bit of a tradition here."
Pence also batted back increased scrutiny of Trump's charitable foundation, with reports emerging that Trump is almost exclusively directing other people's dollars -- not his own -- to charity work.
"Anyone who knows about Donald Trump his career knows that this is a man that has given away tens of millions of dollars to charitable causes," Pence said. "There's simply no question: You can talk to charities all across the country who have benefited by the generosity of the Trump family. And I am very, very proud of their charitable record."
Pence reiterated that he himself believes Obama was born in Hawaii, but said the campaign and country had moved well beyond the Trump-led controversy: "It's a four-year old issue. It's not what the American people are talking about."
Trump during the 2012 campaign was the leading skeptic of Obama's birthplace, insisting that his constitutional legitimacy as a president was in doubt. He has repeatedly declined during the 2016 campaign to either walk back or expand on the claims, arguing that it would draw media coverage and distract from what he considers the race's core issues.
Praise for Putin
Trump and Pence have also caught fire for laudatory comments they've made about Vladimir Putin, the Russian leader who is widely denounced by US politicians of both parties. Both members of the GOP ticket have said that Putin is a stronger leader than Obama, but Pence said that the ticket is not naive about his Russian counterpart.
"Donald Trump and I know what we're dealing with," Pence said, heralding the businessman's skills as a dealmaker. "But what you have in Donald Trump is someone who will take the world as it comes."