Doing a quick review of the literature I can find shows we don't have a completely clear picture yet. The link in young children is definitive and the link in adolescents is murky but it looks like one of many risk factors. I am inclined to entirely disregard the linked study as garbage since it gives a very definitive black-and-white answer that is in opposition to highly cited literature. I think the main reason it is at the top of HN is that we want this to be the simple black-and-white answer.

As someone who grew up with violent video games I think this issue is taken way too personally by most of us. The idea that video games will necessarily make you violent is absurd and not being proposed by anyone (that is mostly a straw man used to dismiss the idea). The question is whether violent video games can push some kids over the edge -- kids who are already predisposed to violence. Imagine an abused kid with no farther figure who would probably be a violent teen anyways. Will playing violent videogames numb him to the idea of killing, give him neural pathways that make the repetitive action more likely and start him with some bad ideas? Could it be the difference between him getting in some fist fights and indiscriminately murdering schoolchildren? The answer looks like "yes."

The case made in the book On Combat is very interesting. That book is considered one of the definitive secondary works on combat psychology and he spends significant time on this question. I strongly recommend the book.