Keith Prince - the chairman of the London Assembly's transport committee - was caught on video by another commuter on a Southeastern service heading into the capital from Dartford, Kent, at around 8.30am on Thursday.
The argument with the young woman - uploaded to Facebook by the Dartford Rail Travellers Association - appeared to have started because she had her feet up on one of the seats and Mr Vince made an attempt to move them.
It quickly escalated and the footage shows her using racial slurs and abusive language.
She describes Mr Prince as "an idiot" and threatens to "f****** slap" him as he attempts to remove her feet from his seat, before descending into a bizarre dispute over the councillor's marriage.
The row continues when Mr Prince makes a further attempt to remove the woman's feet from his seat.
In his apology, he said: "I regret getting involved in such a petty dispute. This is one of those situations in which it is better to simply walk away.
"By the end of the journey we were joking with each other and having a pleasant conversation. That said, I'd like to apologise to my fellow commuters for the disturbance our initial disagreement caused."
Mr Prince began his role at the transport committee - which examines the capital's transport system in order to press for improvements - last year.
He is also the former leader of Redbridge Council, where he remains a councillor.
A Southeastern spokesman said no complaints had been received about his dispute with the woman.