"I am a US senator. I have a prediction -- this is not a guarantee -- and I'm not revealing inside intel," said Kaine, who represents Virginia.
"I was in the Senate when the Republicans' stonewalling around appointments caused the Senate Democratic majority to switch the vote threshold on appointments from 60 to 51. And we did it on everything but a Supreme Court justice."
"If these guys think they're going to stonewall the filling of that vacancy or other vacancies, then a Democratic Senate majority will say, 'We're not going to let you thwart the law.' And so we will change the Senate rules to uphold the law that the court will be nine members."
So far, Senate Republicans have refused to hold hearings for President Barack Obama's nominee, Merrick Garland, following the unexpected death in February of Justice Antonin Scalia. Democratic presidential nominee Hillary Clinton has not indicated whether she plans to renominate Garland or make a different appointment if she wins the White House.
Kaine made his prediction during an interview with The Huffington Post
at a Columbus, Ohio, brewery. The interview was released Friday.
The vice presidential candidate also weighed in on comments by his Senate colleague Ted Cruz suggesting he would work to hold the court vacancy if Clinton wins, leaving the court with eight members.
"There is certainly long historical precedent for a Supreme Court with fewer justices. I would note, just recently, that Justice (Stephen) Breyer observed that the vacancy is not impacting the ability of the court to do its job. That's a debate that we are going to have," Cruz told reporters in Colorado earlier this week, according to The Washington Post.
Kaine called Cruz's comments "disingenuous."
"Ted Cruz I think is being slightly disingenuous. The Constitution doesn't set the size of the court. It sets a maximum, I think, of 15. But since the Judiciary Act of 1869, it's been a nine-member court. That's what the statute says," Kaine said.
"We take an oath to uphold the law, that's what we're upholding. Congress could change it, if Congress wanted to and had the votes, but Congress won't. So until Congress does, that's the size of the court."
Clinton's running mate suggested it was still possible for Garland to get a vote in the Senate.
"After an election, people will think about it, and I think there's still a significant likelihood that Merrick Garland will get a vote before the end of the year," Kaine said.