The original punishment handed down to the former France international was widely condemned as too lenient, but the governing body argue an appeal will not succeed
Anelka was handed a five-match ban and fined £80,000 by an independent three-man panel for performing the gesture in honour of his friend, controversial French comedian Dieudonne M'Bala M'Bala, after scoring in a 3-3 draw with West Ham at Upton Park on December 28.
The chosen punishment by the panel was widely condemned as too lenient, as was their judgement that Anelka is not anti-Semitic, and the FA considered appealing for a harsher sanction.
But the governing body have now decided against such a move, arguing it would be unlikely to succeed.
"Having carefully considered the written reasons of the Independent Regulatory Commission together with our appointed QC, The FA can confirm they will not be appealing the sanction imposed on Nicolas Anelka," a statement on the FA's website read.
"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."
In addition to the sanction handed down by the independent panel, Anelka is also currently suspended by West Brom pending the findings of an internal club investigation into the incident.
During the course of his hearing, Anelka also admitted performing a 'quenelle' after his second goal against the Hammers.