Pete Buttigieg (D), Mayor of South Bend, Indiana, seen speaking at the National Action Network National (NAN) convention in New York City, NY.

Democratic presidential hopeful Pete Buttigieg, continuing his war-of-words with fellow Hoosier Mike Pence, said the vice president could "clear this up" by taking a broad stance against discrimination.

"I'm not interested in feuding with the vice president," Buttigieg said Thursday during a taping of "The Ellen DeGeneres Show."

"But if he wanted to clear this up, he could come out today and say he's changed his mind, that it shouldn't be legal to discriminate against anybody in this country for who they are. That's all."

Buttigieg, the mayor of South Bend, Indiana, worked closely with Pence while Pence was governor. Since formally announcing his exploratory committee in late January, Buttigieg has rocketed to the upper tier of the Democratic presidential race. If successful, he would be the first openly gay nominee of a major party.

The spat with the vice president began last weekend when Buttigieg critiqued Pence's record on LGBT equality during an event hosted by an activist group, and escalated on Wednesday when the vice president accused Buttigieg of attacking his Christian faith during an interview on CNBC.

"I'm not critical of his faith. I'm critical of bad policies," Buttigieg said Thursday. "I don't have a problem with religion. I'm religious, too. I have a problem with religion being used as a justification to harm people and especially in the LGBTQ community."

In an interview that aired Friday with CNBC's John Harwood, Buttigieg described himself as "the only Maltese-American Episcopalian gay veteran that I know."

Pence, in a CNN interview that also Friday, Pence said he considered Buttigieg a friend, and didn't "believe in discrimination against anybody. I treat everybody the way I want to be treated."

"Pete has his convictions, I have mine," Pence said.

Pence has long faced criticism from LGBT activists for his opposition to gay marriage and his support for legislation that his critics said targeted gays and lesbians. Pence has noted that as governor, he worked to implement the Supreme Court decision that made gay marriage the law of the land.