They said the former Liverpool defender had personally contacted the family to apologise and had shown genuine remorse for his actions.
The 40-year-old spat at the car where the young Manchester United fan was travelling in following a brief verbal exchange with the teenager’s father after his former club lost 2-1 to the Red Devils at Old Trafford on Saturday.
Footage filmed by the parent and obtained by the Daily Mirror shows the father tease Carragher about the result as the pair drive alongside each other after the game.
In the video, he shouts through his daughter’s front passenger side window: “2-1, 2-1, unlucky Jamie lad, 2-1 lad.”
Carragher then winds down the window of his black SUV and spits towards the car.
The girl tells her father “he spat on me”, to which he responds: “Jamie Carragher spat at my daughter… nice, nice, very nice Jamie Carragher”.
Speaking to Sky News, the ex-Liverpool defender called it “a moment of madness”.
The football pundit, who has a daughter the same age as the girl who was caught up in the incident, said his behaviour “cannot be condoned” and became visibly upset when he was asked what his children thought of his behaviour.
He said his “biggest regret was that a young girl was caught up in the middle of the altercation”.
“That devastates me more than anything. I’ve got a daughter exactly the same age and I can’t say how I’d react if someone did that to her,” he continued.
“I’ve made mistakes in my 25-year career, but this is the worst one.”
Carragher added: “Hopefully I can get the chance to meet face-to-face with the family, and apologise to them again.”
He was suspended by Sky Sports shortly before the interview, which said in a statement: “Sky takes this matter extremely seriously and strongly condemns Jamie’s actions, we have made that clear to him in person today and suspended him from his duties.
“It falls well below the standards we expect of our people.”
The girl’s father said the family “don’t want him to lose his job”.
He told the Mirror: “We wanted an apology and an explanation. He seems contrite. Everyone makes mistakes, we are all human.”
His wife added: “He did seem extremely sorry.”
Fellow Sky Sports pundit, former Manchester United defender Gary Neville, also urged his bosses not to sack Carragher.
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He tweeted: “I’ve just watched Carra23 say sorry. No excuses he’s made a big mistake. He’s massively passionate about football and he’s overstepped the mark and shouldn’t have reacted.
“I’ve been on TV for 3 years with him and imo this isolated incident shouldn’t stop us working together.”