The woman described the incident as a "horrible moment" that she had had to live with all her life.
In an open letter published on the RadioLive site, Fagan apologised as he "got it so wrong".
"I arrived in the studio exactly one minute before we went on air, extremely annoyed at having been left on the side of the road and when Elle called I was still so wrapped up in what had occurred I was simply not listening and I did not hear what she said. I realise, listening back, that my challenging of her in that context was obnoxious and hurtful."
The apology came after hosts John Tamihere and Willie Jackson announced they would not be returning to air for the rest of the year after an interview with a friend of an alleged victim of the Roast Busters group.
The pair said they "deeply regret" the comments made last week in an interview with 18-year-old Amy.
Tamihere said they would be taking the next few weeks to review what happened last week with management and agree what action needed to be taken.
"We do not condone rape in any way and did not intend to blame the victims."
When asked why it took so long to take the hosts off air, spokeswoman Rachel Lorimer said the announcement was made yesterday, but not the decision. She would not say whether the hosts would be paid while they were off air.
More than 63,000 people have signed an online petition calling for Prime Minister John Key to take action over the Roast Busters.