In a live interview on Sky News on Monday afternoon, Carragher, who called the family on Sunday, said: "It was a moment of madness that is hard to explain".
Carragher who could yet still be sacked from the broadcaster, has been suspended indefinitely by the company after he was caught spitting at a 14 year old girl. All I can do is apologise as much as I can. I have a daughter of the same age and if someone had done that to her I don't know how I would respond.
As a result Carragher was unsurprisingly dropped from the Monday Night Football line up as Stoke took on league leaders Man City.
He tweeted: "I've just watched Carra23 say sorry".
A spoksperson for Sky Sports said: "It is unacceptable behaviour and we will be addressing it with Jamie".
He joined Sky after retiring from playing in 2013 and has won plaudits for his tactical analysis on the flagship "Monday Night Football" show and other Premier League broadcasts.
Carragher made a huge mistake, there are no two ways about it, but he's done the right thing by apologising to the 14-year-old girl and her family at least.
Admitting that his position both as a footballer and a television pundit means he is a role model with responsibilities, Carragher accepted that he has a duty not to act as he did on Saturday evening. "It falls well below the standards we expect of our people", Sky Sports said in a statement. That five seconds will not take over everything that I've done.
"Hopefully they accept that and I'd like to apologise again for that if possible".
"I've been on TV for three years with him and imo [in my opinion] this isolated incident shouldn't stop us working together".
"[It was] a moment of madness that really is hard for me to explain".
"I haven't offered my resignation but they have made it known that they are very disappointed about the shame now hanging over the name".