I'm going to tell my ex's new girlfriend that he cheated on her
Why would you choose to tell her that?
Because she's thinking about betraying my ex, so I think she's an ugly little bitch.
It would be unwise to tell your ex's girlfriend that he cheated, whether your are telling the truth or making it up. He's your ex, so she's unlikely to believe you, and you would be getting involved in their relationship when it's none of your business.
Maybe. You might be right about this.
It is best to just stay out of this. Let things work themselves out and live your life.