Carragher has apologised for the incident, which took place after Liverpool lost 2-1 to Manchester United on Saturday, calling it "a moment of madness".
The 40-year-old former England international was summoned for talks with Sky Sports who had earlier described his actions as "unacceptable".
Carragher was driving home after summarising on Manchester United's win over Liverpool at Old Trafford when he apparently became involved in a conversation with the occupants of an adjacent car.
A video taken from the car, released on the website of several British newspapers, clearly shows Carragher spitting.
In the video the father of the girl can be heard shouting: "Unlucky Jamie lad. Two, one".
"Totally out of order and I've apologised personally to all the family this evening," Carragher said on Twitter.
"I was goaded 3/4 times along the motorway while being filmed and lost my rag. No excuses, apologies."
Carragher, who joined Sky Sports after retiring as a player in 2013, has become a popular pundit working regularly alongside former United defender Gary Neville.
(Reporting by Martyn Herman and Simon Jennings, editing by Ed Osmond)
