"We're like two brothers who were somehow raised by different fathers or something," Biden said of Sen. John McCain, who was diagnosed with the same kind of brain cancer that killed Biden's son, Beau.
Biden moved to sit next to Meghan McCain, saying that her father is seeing one of the doctors his son consulted after being diagnosed with glioblastoma.
Meghan McCain and Biden held hands and both started tearing up when the former vice president talked about the hope that his son kept alive even after getting "this devastating diagnosis, like your dad."
Biden added that Beau, who died in 2015, said "we're not going to talk about percentages" after receiving the diagnosis.
"Beau is missing but we've all decided to not talk about the loss as much as the inspiration he was to all of us," Biden said.
Speaking about the continued speculation about his own future, Biden didn't give a concrete answer about whether or not he will run for president again in 2020, saying that time will tell.
"If I were offered the nomination by the Lord Almighty today, I would say no because we're not ready, the family's not ready," Biden said, but added that if he were asked next year, "I may very well do it."
Earlier in the interview, Biden also shared his excitement about his former colleague Doug Jones' special election win in Alabama this morning.
"This is not just about rejecting Trump -- which was real -- but it's about Doug Jones," Biden said.
"I just think I'd rather focus on the positive side of Doug Jones than the ugly side of that side," Biden said.
Biden said that even if Trump's claim that he speaks about men the same way he speaks about women is true, Biden balked at "the idea that the president is so tone deaf" as to apply that language to a female politician.
"When our leaders use the kinds of language that is so often used today ... we allow the seamier side in society to act as if it has a legitimate place," Biden said.