"I just want you to know I had permission to hug Lonnie," Biden said, referring to International Brotherhood of Electrical Workers male president Lonnie Stephenson, whom he hugged after Stephenson introduced him.
Later, Biden invited a group of children in the audience up to the stage, and while putting his arms around a young boy and talking about the financial future of younger generations, made a similar joke.
"By the way, he gave me permission to touch him," he said to laughter from both the boy and the audience.
Soon after the speech, the woman who first accused Biden of inappropriately touching her denounced his remarks on Twitter.
Biden "hasn’t reflected at all on how his inappropriate and unsolicited touching made women feel uncomfortable," said Lucy Flores, a former Nevada lawmaker who said Biden kissed her on the back of the head while campaigning for her in 2014.
"To make light of something as serious as consent degrades the conversation women everywhere are courageously trying to have," Flores said.
Biden made no other references to the accusations in his speech, but afterward told a group of reporters outside the event that his comments weren't intended to make light of anyone's "discomfort."
"I hope it wasn't taken that way -- but I literally think it is incumbent upon me, and I think everybody else, to make sure that if you embrace someone, if you touch someone, it's with their consent, regardless of your intention, even if you're trying to bring solace or welcome," Biden said, marking a shift in tone from the speech an hour earlier. "It's my responsibility to do that," he said.
"Like for example, what made me say it -- I wasn't joking -- the president of the union put his arms around me. Well, that's how it's always been. I've been coming here for a long time, that's how people react," Biden said.
Biden also said he "wouldn't be surprised" if more women came forward and said they'd felt uncomfortable in his presence. "But I've had hundreds and hundreds of people contact me who I don't know and say the exact opposite," Biden said.
In that video, Biden said he would continue to govern in a way that connects with people but that he would "be more mindful and respectful of people's personal space, and that's a good thing." He added, "I worked my whole life to empower women. I've worked my whole life to prevent abuse ... and so the idea that I can't adjust to the fact that personal space is important, more important than it's ever been, is just not thinkable. I will. I will."
On Friday, Biden was asked if he had a more direct apology for the women making accusations, rather than a general acknowledgement that his actions unintentionally made people uncomfortable.
"I'm sorry I didn't understand them. I'm not sorry for any of my intentions. I'm not sorry for anything I've done," Biden said. "I've never been disrespectful intentionally to a man or a woman. That's not the reputation I've had, since I was in high school for God's sake."
Biden expressed a clear intent to move forward with his plans regarding 2020 despite the accusations, saying he was very close to "making a decision to stand before you all relatively soon."
Though Biden is a front runner in early polling, President Donald Trump has frequently sought to downgrade his potential opponent. Earlier Friday, Trump said he didn't consider him a "threat."
Despite Trump's own history of more than a dozen sexual misconduct allegations, Trump didn't miss the chance to taunt Biden and sarcastically "welcome" him back into the political spotlight, on Thursday tweeting a doctored video with images of a ghostly Biden slowly rising behind the actual Biden's shoulders, peering into the frame.
Asked on Friday if he’s a “good messenger” to wage a fight against Biden given the accusations women have made about him, Trump said he thinks he is and pointed out what he said was the humor in his tweet.
“Yeah, I think I’m a very good messenger,” Trump said to reporters at the White House. “People got a kick out of it. He is going through a situation, let’s see what happens. But people got a kick out of it. We’ve got to sort of smile a little bit, right?”
Biden, who dismissed the president's video on Twitter, denied to comment further Friday, saying only that he wasn't surprised and that "everybody knows who Donald Trump is."
Flores, the first woman to come forward with accusations of inappropriate touching by the former vice president, shared her story publicly last Friday. She described the interaction with Biden to ABC News as "awkward and disturbing."
A few days later, former Democratic congressional aide Amy Lappos said Biden rubbed noses with her at a 2009 fundraiser, an action she said left her feeling uneasy.
"It wasn't sexual I don't think, but it was incredibly uncomfortable, and it was not how he greeted the congressmen," Lappos said.
Two more women told the New York Times on Tuesday that they also felt uncomfortable after interactions they had with the former vice president. One, who was 19 at the time and a sexual assault survivor, said Biden rested his hand on her thigh at an event at the University of Nevada Las Vegas. The other woman, who was 59 at the time, recalled Biden putting his hand on her shoulder before dropping it down her back while taking a photo, leaving her "very uncomfortable."
ABC News' Molly Nagle and Alexander Mallin contributed to this report.