Overton, who allegedly also told 34-year-old Zaidi to "get back to your own country", received a two-game ban.
Zaidi did not hear the alleged comments which were reported by non-striker Michael Vardy and umpire Alex Wharf.
Overton has denied the claims but Zaidi said: "For the sake of the game he should apologise to me."
Somerset have said the player, along with the county's officials, had written an apology to Sussex and it was accepted.
The 21-year-old, who has been part of England's one-day squad and came close to selection for the winter Test tour to South Africa, was found guilty of a level one offence for abusive language.
It was Overton's third offence of the season and took him to over nine penalty points, meaning an automatic two-game suspension was imposed.
However, Yorkshire captain Andrew Gale received a four-match suspension for a similar offence in 2014. Gale was also stopped from lifting the Championship trophy.
The England and Wales Cricket Board have defended their handling of the case.
"I am not in a position to say whether he was treated leniently but I am sure he could have had a harsher punishment," Zaidi told the Daily Telegraph.
Zaidi was released by Sussex at the end of the season.
Pakistan-born ex-Sussex player Ashar Zaidi says Somerset's Craig Overton should apologise after allegedly swearing at him during a match.