While there's no exact quote about an "extended fight" with "Carney" who beat up his brother, the scene you're thinking of involves Cagney's character, Tom Powers, getting into a brawl with a man named Matt Doyle, played by Donald Cook, who had injured his brother "Mike" (played by Robert Armstrong) in a previous scene.
Here's a breakdown of the scene:
* Cagney's character, Tom, spots Matt Doyle in a diner, sitting with his girlfriend.
* The tension is palpable, and Tom approaches Matt with a clear intent to fight.
* The fight breaks out, and it's brutal and intense, with both men landing punches and kicks.
* Cagney's delivery of lines like "You dirty little rat! You've been walking around, and your brother's in the hospital! And you think you can get away with it?" adds to the intensity and ferocity of the scene.
This scene became iconic for Cagney's violent and energetic performance.
It's important to note that while this scene is widely known and often cited, there's no direct quote mentioning a specific "Carney" character and an "extended fight". It's likely you're combining memories of the scene with other gangster movie tropes.