The incident that you are referring to in chapter 4 is the famous tree jumping scene where Gene shakes the branch which causes Finny to fall out and break his leg. It's during this chapter that the reader begins to see an entirely different side to Gene. The reader realizes that Gene is in constant competition with Finny and thinks that Finny actively tries to pull Gene away from opportunities to do well. Later in the chapter Gene learns that Finny does indeed want Gene to do well in whatever he attempts. That scene further cements Finny as an extremely altruistic character. Finny remains that way throughout the novel.
After this chapter there is always doubt surrounding Gene from the other boys at school. They all seem to suspect that Gene bounced the branch on purpose, which eventually comes to a climax at the trial sequence. That trial causes Finny to realize that Gene isn't as altruistic as he is, which causes him to run out, fall down the stairs, re-break his leg, and die during the surgery.
So in a nutshell, the tree incident ultimately causes Finny's death.