The former Arsenal and Real Madrid striker caused a storm in his native France by performing the 'quenelle' salute made famous in his homeland by comedian Dieudonne M'Bala M'Bala after scoring the first of his two goals at Upton Park in the 3-3 draw.
The gesture has been linked to anti-Semitism in France and The European Jewish Congress has demanded action, claiming Anelka should be subjected to the same punishment handed out to those who perform a Nazi salute.
French minister for sport Valerie Fourneyron also condemned the gesture as "shocking" and "disgusting".
But Anelka is adamant it was nothing more than a "special dedication to his friend Dieudonne".
He expanded on Twitter: "I do not know what religion has to do with this story.
"This gesture is a dedication to Dieudonne. With regard to the ministers who give their own interpretations of my quenelle, they are the ones that create confusion and controversy without knowing what the gesture really means.
"I ask people not to be duped by the media.
"And of course I am neither racist nor anti-Semitic."
Dr Moshe Kantor, president of The European Jewish Congress, believes Anelka should be punished, however.
"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."
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."
It is understood 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 international 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.