The Football Association said the Cameroon international, who remains on Tottenham's books but is out of favour, had been warned about his future conduct and would have to complete a compulsory education course.
The 30-year-old had congratulated Anelka for making the gesture, which has links to anti-semitism and landed the then-West Bromwich Albion player a five match ban, during a goal celebration in December 2013.
The Tweet, in French slang, was subsequently deleted. North London side Spurs are known for their strong Jewish support.
Anelka, who recently joined the Indian Super League, denied he was anti-Semitic but was later sacked by West Brom for what they described as gross misconduct.
Hull City striker Yannick Sagbo was fined 15,000 pounds for also expressing support for Anelka via Twitter.
Assou-Ekotto had denied an FA charge of improper conduct but the football body's regulatory commission found he had breached the rules.
However, it added that it was "satisfied that when the player sent the tweet... in his mind he believed he was congratulating Anelka on what he perceived to be an anti-establishment gesture as opposed to one associated with anti-Semitism."
The FA said the sanction was stayed pending a possible appeal, with the player given seven days to respond once he had received a written decision.
The player said in a separate statement issued through a lawyer that he was pleased the FA had acknowledged he was not anti-Semitic.
"I have always loved the fans of Spurs and I want to assure them that I would never knowingly behave in a racist or anti-Semitic manner," he said.
(Reporting by Alan Baldwin, editing by Martyn Herman)
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