The West Brom striker performed the salute after scoring in December's 3-3 draw at West Ham in the Premier League and was charged by the FA in January.
Anelka denied the charge and defended the gesture by saying that it was "anti-system" and a dedication to his friend and comedian Dieudonne M'bala M'bala as opposed to being anti-Semitic, but the FA took a dim view of the incident.
An independent tribunal subsequently handed the Frenchman a five-match ban and last Thursday published the reasons for its findings.
It revealed that the FA had been seeking a longer ban, based on the fact that Anelka denied the charge, is a prominent figure and his actions undermined the governing body's stance on discrimination of any kind.
However, the FA have now opted against appealing the punishment, saying in a statement: "Having carefully considered the written reasons of the Independent Regulatory Commission together with our appointed QC, The FA can confirm it will not be appealing the sanction imposed on Nicolas Anelka.
"The grounds of appeal available to us are limited to legal challenges or to circumstances in which the sanction imposed is "so unduly lenient as to be unreasonable ".
"As the written reasons make clear, The FA pressed the Commission at the original hearing to impose a suspension of greater than five matches.
"We do not consider there is a real prospect of successfully appealing to extend the sanction imposed."
Anelka, who is currently serving a club suspension, has until the end of Thursday to lodge an appeal of his own. If no appeal is forthcoming, the former France forward will serve the first instalment of his five-match ban against Swansea City on Saturday.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.