Gene is absolutely guilty of causing Finny to fall from the jumping tree. His actions caused the branch to move and threw Finny off balance to such an extent that even his natural athletic abilities were not able to compensate.
Of course, Gene didn't interpret the situation from this viewpoint at the time it happened. Gene had gone in the course of one conversation from feeling irritated and angry at Finny for causing continuous distractions when Gene wanted to study to feeling overwhelmed with the realization that Finny sincerely wanted Gene to do well in his studies, that there was no sense of competition between them. Gene could not bear to admit to himself that Finny was that generous and open with their friendship. The jouncing of the tree branch was not premeditated, but it certainly was Gene's fault.