Does violent media cause people to act violently? No.
Does violent media contribute to people acting violently? Probably.
Compare, for the sake of argument, two extremes:
1. A child raised in an environment with constant exposure to movies and video games that depict violence, gore, murder, etc. (Consider also that the video game is not as passive, like a movie—you are actively rewarded for committing acts of imagined violence.)
2. A child raised in an environment with no exposure whatsoever to similar violence.
I think it is reasonable to assume that, all other things being equal, child 1 would be less horrified by real-world violence, more accepting of real-world violence, and possibly, when faced with an extreme situation, more inclined to use real-world violence.
But I hate all this guesswork. So here’s what the science says (and note, these were not cherry-picked—these are the top hits from google scholar):
“Adolescents who expose themselves to greater amounts of video game violence were more hostile, reported getting into arguments with teachers more frequently, were more likely to be involved in physical fights, and performed more poorly in school.”
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WH0–4BF0D7T-1&_user=4422&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059600&_version=1&_urlVersion=0&_userid=4422&md5=e35f7bc00ea859c1f652967416613d06
“Research on exposure to television and movie violence suggests that playing violent video games will increase aggressive behavior. A meta-analytic review of the video-game research literature reveals that violent video games increase aggressive behavior in children and young adults. Experimental and nonexperimental studies with males and females in laboratory and field settings support this conclusion. Analyses also reveal that exposure to violent video games increases physiological arousal and aggression-related thoughts and feelings. Playing violent video games also decreases prosocial behavior.”
http://www3.interscience.wiley.com/journal/118998785/abstract?CRETRY=1&SRETRY=0
”...all the published studies on video game violence have methodological problems and that they only include possible short-term measures of aggressive consequences. The one consistent finding is that the majority of the studies on very young children—as opposed to those in their teens upwards—tend to show that children do become more aggressive after either playing or watching a violent video game.”
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VH7–3V8C7YH-6&_user=4422&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=4422&md5=604bf1d11628da129dfc295b8b9ed96b
“An updated meta-analysis reveals that exposure to violent video games is significantly linked to increases in aggressive behaviour, aggressive cognition, aggressive affect, and cardiovascular arousal, and to decreases in helping behaviour. Experimental studies reveal this linkage to be causal. Correlational studies reveal a linkage to serious, real-world types of aggression. Methodologically weaker studies yielded smaller effect sizes than methodologically stronger studies, suggesting that previous meta-analytic studies of violent video games underestimate the true magnitude of observed deleterious effects on behaviour, cognition, and affect.”
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WH0–4B9D74R-1&_user=4422&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059600&_version=1&_urlVersion=0&_userid=4422&md5=fa3db762bcc9498f1af2b7ff839223e6