That's a rare policy disagreement with Clinton, the Democratic presidential nominee, and her running mate, who said he was "so humbled" to be chosen for the ticket.
"My job is really to just do everything I can to support a great Clinton presidency and I'm really excited to be off and running on that task," the Virginia senator said.
Abortion, though, remains a point of difference between the pair. Kaine said he supports the Hyde Amendment, a 40-year-old rule preventing federal taxpayer dollars from funding abortions. That contradicts comments by Clinton campaign manager Robby Mook in a July 24 "State of the Union" appearance. Kaine "has said that he will stand with Secretary Clinton to defend a woman's right to choose, to repeal the Hyde Amendment," Mook quoted Kaine as saying.
"My voting position on abortion hasn't really changed," Kaine said in the interview aired Sunday. "I support the Hyde Amendment. I haven't changed that."
Tapper pressed Kaine, saying Mook told CNN otherwise.
"That is not accurate and I don't think Robby has said that, Jake," Kaine responded.
Clinton spokesperson Jesse Ferguson told CNN on July 26 that Kaine's commitment to stand with Clinton on repealing the Hyde Amendment "was made privately."
Kaine said in the interview that he thought about his differences with Clinton over abortion before joining the ticket. As a potential vice president "I had to get comfortable with the notion that I can have my personal views but I'm going to support the president of the United States, and I will."
Still the issue is likely to linger among some Clinton supporters. NARAL Pro-Choice America president Ilyse Hogue called Kaine's continued opposition to repealing the Hyde Amendment "deeply disappointing" in a tweeted statement that was apparently deleted and then reposted
"While we appreciate Senator Kaine's clarification that he will support the nominee's position on this, we sincerely hope that Sen. Kaine will continue to educate himself on what Hyde means to the most vulnerable women in this country and join us in fighting this injustice," the statement said.
On Sunday the group tamped down its criticism, and tweeted it is now "glad" Kaine will stand with Clinton to "end Hyde," exhibiting a more optimistic outlook on Kaine supporting Clinton regardless of his personal views.
Tapper asked Kaine if he agreed with former New York City Mayor Michael Bloomberg's suggestion in his speech at the Democratic convention that Trump is not sane.
"I don't know anything about his sanity. I haven't met him," Kaine said. But he did express confusion at some of Trump's statements and tactics, especially a recent episode in which Trump confused Kaine with a former Republican governor of New Jersey, Tom Kean.
"Tom Kean was governor of New Jersey and he finished his term 26 years ago," Kaine said. "I've got to scratch my head about that guy every day."
Kaine, a former Virginia governor, also criticized Trump's seeming invitation last week for Russia to hack into US servers and find Clinton's missing emails -- a comment Trump later said was "sarcastic."
"I don't think cyber security is a big joking matter," Kaine told Tapper. "The guy's got a bizarre sense of humor."
Unifying the Democratic party
Kaine said that, "sure, there is" work to be done to unify the Democratic party. "But I feel like we're in really good shape."
The convention in Philadelphia got off a to a rocky start due to protests by Bernie Sanders supporters angry at what they called bias toward Clinton's campaign by the Democratic National Committee. The protests came after the group WikiLeaks released thousands of hacked emails portraying DNC staffers in an unfavorable light.
Kaine said these kind of arguing factions make the Democratic party better than their Republican rivals.
"I'm used to the fact that we're a big family with a lot of different points of view. Unlike the Republicans, you know we haven't had the litmus test where if you don't believe this, we're going to throw you out. We have ideological breadth and that's good, that's what we want to have."
Kaine, an early backer of Barack Obama in the 2008 in his primary fight against Hillary Clinton, said he believes that the Democratic Party is more unified today than it was at this point eight years ago, following the protracted primary battle between the pair.
On Hillary Clinton making history
The Democratic National Convention made history on Wednesday by nominating Clinton, a former secretary of state, as the first female presidential candidate of a major party. Kaine was quick to share his own awe, as well as his mother's.
"I said to Hillary when I walked out, when I walked out into that fantastic evening I said to her, 'This is a great country, and you just made it a lot greater.'"
Kaine's mother, he recalled, "said, 'This is the best night of my life.' Now when mom says that to her son I feel like I know at least for the next 48 or 72 hours, I'm going to be in good shape."