The other driver, who filmed the footage, repeatedly shouted the 2-1 score to Carragher, who is heard to say, "All right", before winding down his window and spitting towards the other vehicle.
Reports said that some of Carragher's spit hit the face of the girl, who was sitting in the passenger seat.
Speaking to Alan Brazil on talkSPORT radio, Jones said: "It is cowardly".
"All I can do now is apologise, I can't go back". Driving away from Old Trafford the family of United supporters spotted Carragher in his vehicle next to them in traffic.
"But to be honest that is where again I think to myself "why did you react like that" because that's part of being a public figure at times, different things get said but you don't react like that". That devastates me a little bit more than anything else. "This whole sorry incident I've dragged that family into is devastating for them, devastating for my family and it's all down to me".
SH: Tell me about the family.
In the video obtained by the Mirror, Liverpudlian Carragher exchanges waves with the girl's dad.
Speaking to the BBC on Monday, he said: "It looks bad and I accept that". I apologise for it.
Carragher has become a staple of Sky's coverage of the Premier League since his retirement in 2013.
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Carragher was due to be working for Danish channel TV3 Sport to cover the match between Manchester United and Sevilla. "He's massively passionate about football and he's overstepped the mark and shouldn't have reacted".