Video footage shows Carragher leaning forward and spitting towards the man's car after the driver said: "Two-one mate, unlucky Jamie lad."
The incident provoked controversy after the man's teenage daughter appeared to have been hit by Carragher's spittle.
"I lost my head," Carragher told The Mirror on Sunday night. "It was a bit more than 'hiya Jamie, it was 2-1'. It went on for two or three times. I drove away at first and it just continued and continued.
"I was leaving the ground and trying to keep my head down. I thought it was a Liverpool fan at first wanting an autograph at first so I stopped. It carried on and I drove away but it carried on again and I lost my head.
"I shouldn't have done it but I was thinking - what is a grown man doing, carrying on like that two or three times with his daughter in the car?"
Carragher, who also issued an apologetic statement on Twitter on Sunday, added: "I will apologise to the daughter."
Totally out of order & Ive apologised personally to all the family this evening. I was goaded 3/4 times along the motorway while being filmed & lost my rag. No excuse apologies. https://t.co/ofrNfiwhYH— Jamie Carragher (@Carra23) 11 March 2018
Sky Sports, which employs Carragher as a pundit, described the incident as "unacceptable behaviour" and confirmed that it will be having a word with the former player about what happened.