In scoring his first of two goals in Saturday's 3-3 draw at Upton Park, the 34-year-old seemingly performed the 'quenelle' salute made famous in his homeland by comedian Dieudonne M'Bala M'Bala.
The gesture has been linked to anti-Semitism in France but Anelka took to Twitter to defend his actions, saying it was "just a special dedication to my comedian friend Dieudonne".
The salute, though, is the subject of a fierce national debate and French minister for sport Valerie Fourneyron condemned the gesture as "shocking" and "disgusting".
European Jewish Congress president Dr Moshe Kantor wants Anelka to be banned as if he "had made the infamous outstretched arm salute" of the Nazis.
"It is sickening that such a well-known footballer would make such an abusive and hateful gesture in front of tens of thousands of spectators," Kantor said.
"There should be no room for such intolerance and racism in sports and we expect that the English Premier League officials as well as the police will give Anelka the appropriate punishment.
"We know that English football officials have a very low tolerance for racism at football matches and we hope that all concerned will abhor and show zero-tolerance for this hateful Nazi salute."
Kantor, describing the 'quenelle' as "merely a lesser known Nazi salute", added: "Furthermore, we hope that this gesture is banned in all places of Europe where Nazi salutes are banned.
"Merely inverting the traditional Nazi salute should not allow anti-Semites to spread and display their hate with impunity."
Anelka's position differs starkly with how the European Jewish Congress views the gesture, and West Brom caretaker manager Keith Downing played down the prospect of disciplinary measures immediately after the match.
Downing said of Anelka: "He is totally unaware of what the problems were or the speculation that has been thrown around, he is totally surprised by it."
Press Association Sport understands the Football Association will be looking into the matter, while anti-racism campaign group Kick It Out has offered its assistance to the national governing body.
Earlier this month FIFA handed Croatia international Josip Simunic a 10-match ban for leading fascist chanting following his team's World Cup play-off victory over Iceland. The defender was captured leading songs with associations to Croatia's former pro-fascist regime.