The Hammers were held to a 2-2 draw away to their London rivals and Cole was agitated when he responded to a question on Twitter.
Spurs fan Stuart Hardy tweeted: "Hi @CarltonCole1 when your own team-mates don't kick the ball out when you're lying injured for 2 mins, you think it's time to call it a day?"
Cole retaliated by telling the fan to 'F off you c***.'
Cole's tweet was eventually deleted, but was subsequently retweeted thousands of times by his followers.
Hi @CarltonCole1 when your own team mates don't kick the ball out when you're lying injured for 2 mins,you think it's time to call it a day?
He had until 6pm on Thursday to respond to the charge but has accepted it and has duly been slapped with a £20,000 fine.
It's not the first time Cole has found himself in trouble with the footballing authorities for comments made on social media, and breached the rules during England's match against Ghana in 2011.
He tweeted: "Immigration has surrounded the Wembley premises! I knew it was a trap!
"The only way to get out safely is to wear an England jersey and paint your face w/ the St George's flag!"
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