Biden, who served eight years as Obama's vice president after 36 years in the U.S. Senate, said he did not have second thoughts, however, about the reasons he passed on the race.
"I don't regret the decision I made because it was the right decision for my family," he said.
Joe Biden explained that in order for people to decide whether they will run for president, they need to be able to answer two questions.
"One, do they truly believe they are the most qualified person for that moment? I believed I was," Biden said. "But was I prepared to be able to give my whole heart, my whole soul and all my attention to the endeavor?
"I knew I wasn't."
Reflecting on the tragedies that have befallen him personally as he wrote his forthcoming book, "Promise Me, Dad: A Year of Hope, Hardship and Purpose," Biden said he recalled a conversation he had with his mother, Catherine "Jean" Finnegan Biden, just after the deaths of his wife and daughter, in which she encouraged him to persevere.
"She said, 'Joey, grab my hand ... Out of everything horrible, something good will come if you look hard enough for it,'" he told Winfrey. "That was my mother's notion. We were taught just to get up. When you get knocked down, just get up and move forward."
Many of those early lessons came during Biden's childhood in Scranton, Pennsylvania, a blue-collar city he mentioned frequently on the campaign trail in 2008 and 2012. Winfrey noted that in those days, he was known as "a boy with a vision" who knew from a young age the kind of person he wanted to grow up to be. Asked if he fulfilled his vision, Biden said yes but said it wasn't a matter of his professional achievements.
"I wanted to live up to my parent's expectations, and I wanted to be that person that met my mother's standard, being defined by my courage," he said. "I wanted to be that person who, no matter what happened, just got back up and kept going. I wanted to be that person who was there and loyal to people who were loyal to him."
Though Biden did not directly refer to Trump in the clips aired by "GMA," he described for Winfrey what he believes makes for successful leaders.
"They understand their strengths, and they understand their weaknesses. They play to their strengths and try to shore up their weaknesses," he said. "And the people who don't do that are the people who aren't self-aware enough to know ... because most of the time that abuse ends up in their downfall as well."
"Super Soul Sunday" with Joe Biden airs Sunday, Nov. 12, at 11 a.m. ET/PT on OWN: Oprah Winfrey Network.