The next day Biden received a call from Kennedy, who joked about the seizure and then asked in a very serious tone how Biden's granddaughter was doing.
"What do you mean, Teddy?" Biden said.
"I'm afraid I frightened her. Is she OK?" he said. Biden's granddaughter, Naomi, then 15 years old, had been sitting at Kennedy's table during the inauguration luncheon where he had the seizure.
It's a story Biden recounted to "Good Morning America" Wednesday just hours after Kennedy's passing, as one example of what Biden called an "incredible quality" that, in part, defined the "Liberal Lion's" life.
"Here is a guy who has a terminal disease, has a seizure in front of my granddaughter and he's only worried about did it negatively affect my granddaughter," Biden said. "There are just so many little things that Teddy did that were just amazing."
Biden said Kennedy was a "constant" factor in his life for more than three decades and credits Kennedy with "helping [him] get elected."
"At every single important stage of my life, he was there. ... He was there when I lost my wife and daughter. He was there when I was ill. He was there at every high point and low point of my life," Biden said. "He felt like a big brother. He was always there for me.
"It just leaves a void. I'm sure for the hundreds of people who knew him well, millions of people wanted to know him better," he said.
In full view of Kennedy's life, including the controversial night in 1969 on the island of Chappaquiddick when a young Ted Kennedy left the scene of a car crash that claimed the life of a 28-year-old woman, Biden said Kennedy did more than outlive his flaws.
"He made up for them," Biden said. "Teddy Kennedy constantly improved. Teddy Kennedy constantly got better. ... Teddy's enthusiasm for life and for the opportunity to make things better. I sat with him for 36 years. Every day and every year he grew. It was a greater amount of passion, not less.
"Every mistake he made in his life, he made a lie out of the mistake by the way he lived the rest of his life," Biden said.
Biden: Those Who Spoke With Kennedy Came Away Acting 'Bigger'
Those who knew Kennedy, even his foes, were better for it, Biden said.
"When he spoke to you, it was never about him, it was always about you," Biden said. "Whenever he debated, he was never petty. He was never small. And, as a consequence, he had his foes, not just his allies, act bigger, more graceful, with more dignity than they ordinarily did.
"How in God's name could you explain the fact that people who had no ideological agreement with him walked away always feeling better for it, always feeling more respected?" he said.
Such a unique quality is something not likely to be seen again for generations, Biden added.
"One, he had the aura and the mystique and all the power of being a Kennedy. Number two, he was totally, thoroughly committed to other people. Everything he did in his public life was about helping someone else. It was real, it was genuine, it was not staged, it was not political," the vice president said. "And thirdly, he had more experience than anyone has had in modern history.
"Those things combined -- the stature, the aura and the commitment, I don't think you'll ever be able to find that again, clearly in my lifetime and I suspect in the lifetime of my children and grandchildren," Biden said.
Biden talked to Kennedy on the phone a few weeks ago, he said, but he was not able to see Kennedy when he went to Eunice Kennedy Shriver's funeral earlier this month. Teddy was having a "bad day."
"So I never got to see him again. But, as Vicki [Kennedy, the senator's wife] said, he was ready to go. But, she said, we were not ready to let him go. None of us were," Biden said. "We relied on him so much. It's almost selfish how those of us who loved him -- and many of us did -- didn't want to let him go.
"He was one hell of a man," he said.