In the interview, Edwards denied knowing anything about any support being provided to Hunter or to Andrew Young, the former Edwards campaign aide who has said that he, not Edwards, is the child's father.
If any such support was being provided, Edwards insisted, "I had nothing to do with any money being paid, and no knowledge of any money being paid, and if something was paid, it wasn't being paid on my behalf."
Edwards' 2008 national finance chairman Fred Baron said late Friday that he had provided "assistance" to Hunter and others without Edwards' knowledge.
In a statement emailed to ABC News, Baron said that he "decided independently to help two friends and former colleagues rebuild their lives when harassment by supermarket tabloids made it impossible for them to move forward on their own."
"I did this of my own volition without the instruction or suggestion of anyone, and made a conscious decision not to tell anyone, including John Edwards, that assistance was provided. The assistance was offered and accepted without condition. This is now and shall always remain a private matter between these individuals and me."
Edwards said he knows he could not be the father based on the timing of the baby's birth on February 27, 2008.
A grainy photo that appears to show Edwards holding a child was published recently in the National Enquirer, which claimed that the baby was Rielle Hunter's child Frances Quinn.
Edwards said he couldn't make "sense" of the photo.
"I don't know if that picture is me," Edwards said. "It could well be. It looks like me. I don't know who that baby is. I have no idea what that picture is."
Pressed to clarify whether the child in the picture was Hunter's child, Edwards suggested that the tabloid may have digitally altered the picture, but also noted the frequency with which politicians hold babies.
"Do you know how many pictures have been taken of me holding children in the last three years? I mean it happens all the time."
Edwards was asked again about the meeting.
"Are you saying you don't remember holding that child of Miss Hunter?"
"I'm saying you asked me about that photograph. I don't know anything about that photograph, I don't know who that baby is. I don't know if the picture has been altered, manufactured, if it's a picture of me taken some other time, holding another baby…I have no idea. I was not at this meeting holding a child for my photograph to be taken I can tell you that."
Edwards finally told Woodruff that he did not remember a baby being present at the meeting last month in California.
Edwards blamed the affair on the adulation surrounding his remarkable rise into presidential politics.
"I went from being a senator, a young senator to being considered for vice president, running for president, being a vice presidential candidate and becoming a national public figure. All of which fed a self-focus, an egotism, a narcissism that leads you to believe that you can do whatever you want. You're invincible. And there will be no consequences."