Clarke, 20, received the offensive online comments on his social networking account from Sky Blues fans after conceding a late penalty in Tuesday night's 2-1 League One defeat at Brentford.
Clarke told the club's official website: "I can take criticism when it's about football. When you play professional football or you are in the public eye, that's what happens and you have to take the rough with the smooth from that point of view.
"But there is just no place in football or society for racism, it's as simple as that."
Coventry have backed Clarke and slammed the remarks.
Development director Steve Waggott added: "I think most players accept that if they engage in social media there are both positives and negatives.
"While criticising an individual player's performance in a match is not particularly helpful or constructive, we cannot stop that.
"However, in these instances, they went way past the mark and were no longer criticism of a player or performance but out-and-out abuse and I am sure every right-minded Coventry City supporter would condemn such behaviour."