Det Supt Fulcher had persuaded Halliwell, 48, from Swindon, not only to confess to Miss O'Callaghan's murder but also to that of Miss Godden-Edwards.
Halliwell was charged with both murders but pleaded not guilty and a High Court judge ruled that the confessions the father-of-three made during a three-hour period on the day of his arrest were inadmissible.
This meant Wiltshire Police had no other evidence against Halliwell to link him to Miss Godden-Edwards' murder and the charge was withdrawn.
Miss Halliwell, 20, said of her father: "He needs to give the other family closure as well and give them the justice they need for their daughter."
Speaking on ITV's Daybreak programme, she said she had received letters from her father but had not replied to them.
He had urged her to change her name a number of times, she said.
"We have had quite a few letters that have said 'sorry for putting you all through this' and things like this," she said. "As much as I want to hate him, I can't.
"I pity him and I do feel sorry for him and I hate what he has done but I can't hate him."
Miss Halliwell said her father had never tried to explain what had happened.
"He said to me - because I saw him - he said 'don't ask me why because I don't know why.'
"So I don't think he even has an explanation," she said.
Miss Godden-Edwards' father, John Godden, has hit out at police, saying they had "made massive mistakes".
He told BBC News: "It seems to me - come to Swindon, commit murder and you'll get away with it.
"And that's the way I feel. I'll never put my trust in the police again. Why should we pay with this pain for somebody else's mistakes?
"I want proper closure. I want closure. I want justice."
He also argued that Halliwell must have had an accomplice.
"Where Becky was found - there was no way he did that on his own," he told The Sun. "Police had to dig her up with mechanical diggers. There is no way he dug a hole that big on his own."
Miss O'Callaghan, 22, disappeared after leaving Swindon's Suju nightclub in the early hours of March 19 last year after a night out with friends.
On the night she vanished, Halliwell had signed off from work but, instead of going home, he cruised the town's streets in his green Toyota Avensis, looking for a victim.
As she made the short walk to the home she shared with her boyfriend, Kevin Reape, Miss O'Callaghan fell into Halliwell's clutches by getting into his taxi.
Police believe Halliwell, who did not know the young woman, took her to Savernake Forest, where he murdered her.
While Miss O'Callaghan's disappearance made national headlines and hundreds of people volunteered to help search Savernake Forest, Halliwell told a work colleague at Five Star taxis: "Who knows what or who you find buried out there? There could be loads of people over the years."
Police reportedly said he had undoubtedly killed other women in a quest to be a serial killer.
A source told the Daily Mirror: "Everything we know about multiple murderers and Halliwell in particular tells us it is inconceivable there was an eight-year gap between the murder of Becky and Sian with no further crimes."
Detectives are expected to investigate the death of Melanie Hall, a 25-year-old hospital worker who went missing from Cadillacs nightclub in Bath in 1996.
