In addition to the tree incident, there are a number of scenes in the book where Gene is not exactly a character of virtue. One that comes to mind immediately is the episode in Chapter 10, when he is visiting with Leper. Leper has gone AWOL, and is hiding out at home, his mind unhinged by his experiences in the military. Gene is shocked that Leper, who had always been polite and reserved, seems to have lost that quality, and now bluntly says whatever is on his mind. When Leper insinuates that Gene is "a savage underneath," and comes right out and accuses him of intentionally causing Finny's accident, his words come too close to the truth, and Gene reacts with violence. He "shove(s) his foot against the rung of (Leper's) chair," knocking the hysterical boy to the floor.
Another incident when Gene is not a character of virtue is when he accompanies Finny to the beach, even though it is against the rules, and he has to study for a test. Gene shows weakness of character in giving in to Finny's insistence, even though he does not want to go. Angry at himself for not standing up for what he thinks he should do, Gene spends the entire trip nursing feelings of bitterness and anger towards Finny, which also is not a virtuous way of behaving (Chapter 3).
There most certainly are. Take for example the incident in chapter seven when Brinker confronts Gene in the broiler room. Brinker starts questioning and hassling Gene over the tree incident. Gene gets super uncomfortable, tries to lie about what happened, and in then, to take the focus off of him, he lashes out at another kid, humiliating him. This is not a very virtuous thing to do; he publicly humiliated some innocent bystander just so that Brinker and the other guys would stop bothering him.
Gene also displays quite a bit of selfishness and bitterness in the first few chapters, against Finny. After Finny confesses his friendship to him on the beach, Gene is silent. Then, instead of thinking, "Hey, that was a really nice thing of Finny to say," he instead decides that Finny did it on purpose to trick him, because Finny was "trying to undermine" his studies. Rather petty and unvirtuous if you ask me.
Then, take Gene's behavior with Leper after Leper is discharged from the army. Leper is in desparate need of friendship and support, and unloads on Gene. He's been through a traumatic experience, is doubting his sanity, and is super vulnerable. But instead of being a good friend to Leper, Gene freaks out on him, yells at him, calls him names, and runs away. Gene behaved very poorly and was an awful friend to Leper.
I hope that those thoughts helped; good luck!