Gene was jealous of Finny before the incident occurred, and to have Finny do yet another spectacular, praisworthy thing just made Gene more envious. Finny was super popular with kids and adults alike, got away with things that no one else could, had a natural affinity for sports and all things cool, and Gene envied him for all of these things. And, to top it all off, he was friends with Gene, which in a way, made Gene feel like he was indebted to him for some reason. So, after Finny saves his life, he feels even more indebted, Finny has yet one more thing on him that makes him feel worthless, and so he reacts with bitterness not gratitude.
Another reason that Gene isn't grateful is that when he was up on that tree limb, he was afraid and terrified. He was mad that Finny had pressured him to be up there in the first place, and him being up there showed himself and everyone else just what a scaredy cat he was. It made Gene look like a baby to have to be saved by Finny, and Gene, already insecure, didn't like that. He states at the beginning of chapter three,
"Yes, he had practically saved my life. He had also practically lost if for me. I wouldn't have been on that damn limb except for him."
So, he feels stupid that he had to be rescued, by Finny the super-kid of all people, and feels a bit abashed at his fear and cowardice on the limb. I hope that those thoughts helped a bit; good luck!