The England wing was suspended for unsportsmanlike conduct after being found guilty of dragging Leicester's Alesana Tuilagi off the field by his hair.
The incident, which occurred in the first half of Leicester's 30-25 Aviva Premiership victory at Welford Road, sparked a touchline brawl and led to Tuilagi and Northampton forward Tom Wood being sent off.
Northampton still disagree with the severity of Ashton's ban but announced on Friday they would not be taking up their right of appeal.
The club said in a statement: "Following considerable deliberation, the Northampton Saints team management has decided not to appeal against the sanction imposed upon Chris Ashton earlier this week.
"Although the management still believes the sanction was harsh, and imposed because of what happened afterwards rather than the action offence itself - something Judge Jeff Blackett noted in his own comments on Tuesday evening - the management also believes that it is best that a line is now drawn under the incident.
"Chris himself is determined to use this as a learning experience and to work hard over the next four weeks to become a better player for Northampton Saints and hopefully earn selection in England's Six Nations squad in the new year."
Ashton will miss Northampton's back-to-back Heineken Cup fixtures against Castres, the Christmas Eve Premiership clash with Bath and the New Year's Day trip to Newcastle.
The England wing will be available for Northampton again in time for the Friday night Premiership game against Harlequins on January 6.
With no specific provision for hair-pulling in the disciplinary code, the three-man panel, chaired by Blackett, equated it with spitting.
Ashton denied the charge and the panel accepted the incident had "started by accident" but Blackett added: "It became clear that at some stage Chris Ashton knew what he was doing.
"If you pull somebody by the hair, there is an element of denigration and this act was a catalyst for a mass brawl that resulted in two red cards and significantly affected the image of the game."
Ashton was also ordered to pay £500 in costs.
Tuilagi and Wood received no further punishment following their red cards.
Leicester duo Geordan Murphy and Horacio Agulla and Northampton lock Courtney Lawes were effectively issued with retrospective yellow cards for their role in the mass brawl.
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