My gut reaction is to say that if "game" referred to sports game, then dump the asshole, but if it referred to Red Dead Redemption or something equally appropriate, then back off my boy, bitch. But upon further reflection, it occurs to me that my gut can be a tad biased.
Sure, your boyfriend sounds like a douchebag. Or at least, contacting someone to get them to contact you so you can ignore them is certainly something a sentient bag of douche might do. But that's only your side of the story, and I'll try and take into account the natural exaggerations inherent to both online communications and passionate talk about a partner.
The more pertinent point here, as I see it, is that you're asking me if you were "right to get mad," as if that were an externally verifiable thing, and my answer would make a difference in your relationship one way or the other. That says something about the way you think of the relationship, and what it's saying you might want to listen to.
It can be the hardest thing in the world to remember, but believe me, dating is not on a point-based system, nor does turning it into a competition guarantee a winner. Make it about who was right to be mad about what, and all you'll end up with is a "participation" trophy stuffed in the abandoned attic of your heart and a lame pizza party in the flooded basement of your regret (apparently some pipes burst in your regret basement).
Honestly, this one can be handled with a conversation. Just make sure to initiate said conversation at a time he doesn't have access to anything distracting, or he just might get wrapped up in a flash game or something and never fini