Here's an idea. After he battles you before the Elite Four (and you win) he says:
"You know what? It doesn't matter! You'll never have the courage to stand up and really be as strong as me! Trainers like you are weak. You'll see, I'm going to be a champion one day, while you're bogged down by your precious Pokémon!"
Then he's the Viridian Gym leader:
"I told you I'd be here. Trainers like you are pathetic. Smothering your Pokémon with love is sick. If you weren't busy back in Johto with that moron professor, you could own a gym, too! But too late! I'll show everyone I can run a gym better than that other guy!"
This would imply that he doesn't know that Giovanni is his father, and Giovanni died. His son took up his mantle without knowing. Then where'd Green end up? Perhaps lingering around somewhere, or wandering Kanto.
I know none of this is very feasible. Just a fun little idea.