I think that the answer to this is yes, and no. I am not sure that Gene actually wanted to cause physical harm. It was, as he stated, "a blind impulse". We have all experienced blind impulses, where we reach out and do something without thinking about it beforehand. We may regret it afterwards, and are puzzled as to why we did it. So, I believe that to a certain extent, his actual jouncing of the limb might have been one of those situations-his body just did it.
However, Gene was excruciatingly jealous of Finny. As a result, he found sinister intentions in everything that Finny did. Any time that Finny took him away from his studies to go play games, there was some resentment, and Gene even concludes later on that "Finny had deliberately set out to wreck my studies." When Finny saved him from falling from the tree, Gene even found malintent in that:
"He had...practically lost [my life] for me. I wouldn't have been on that damn limb except for him."
Even in Finny's selfless act of steadying Gene in the tree, Gene felt he was being selfish. So, Gene has bitter reserves built up, based on jealousy and envy. I am sure that some of those reserves drove his action on the tree that day. There had to be some of that bitterness behind the instinctual action that occurred. In his mind, he had reason to do it, even if he didn't expect the disasterous results, just as someone in anger lashes out and says or does things they regret later on. In the end, I feel that there was a bit of both-the initial action was spurned by Gene's anger, but it happened before his rational filter could stop it from happening.
Gene may not have consciously wanted to harm Finny by causing his fall from the tree, but there are hints that cause us as readers to infer that Gene subconsciously wanted Finny to sustain the injuries he did during the incident.
For one, there is a great amount of envy on the part of Gene toward Phineas. Finny is outgoing, witty, fun to be around, and extroverted, all traits that Gene wishes he possessed to a greater degree.
Also bubbling beneath the surface of the story is Gene's outright resentment of Finny for taking him away from his academics by way of peer pressure. He blames Finny for any of his oversights in studying, as Phineas is always the one who convinces Gene to go play or hang around in places and activities that distract him from his actual work.
These factors put together provide us as readers with the impression that Gene might have had a subconscious motive to jounce the tree limb that caused Finny's fall in the first place.