Under enough duress, many people will falsely confess to things they did not do just to get the questions to stop. They believe the only "right" answer that will alleviate their extreme stress of the moment is to give the interviewer what he or she wants, even if it is not true -- confirmation that they engaged in behavior that they're being accused of and questioned aggressively about.
You're requesting help in proving to him that you're innocent, but his mind is made up. If you were to reconcile he would always hang this over your head -- your lying, the questionable status of the cheating. Don't go back.
You should be more worried that he engaged in such demanding and aggressive questioning that you felt so powerless and desperate that you falsely admitted to cheating just to escape the situation. What right did he have to badger you like this? He did not just challenge your integrity by alleging you cheated. He BULLIED you into a false confession. Be happy you are rid of this guy. It's emotionally ABUSIVE behavior. Down the line, you'd be seeing more than the recent verbal aggression, jealousy, and angry accusations. What he demonstrated should be as deal-breaking as if he slapped you.
Sweet girl, stop your ex-boyfriend's blame game. Break off all communication with him for good. It's better to be alone than be treated like this. Demand to be treated with respect and dignity in all personal interactions.