I'm not really sure how this violates the First Amendment, to be honest.
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."
We can ignore the religion part, because that has nothing to do with this.
The people can still assemble if they so wish.
Freedom of speech is, I'm guessing, the issue here. Now this really depends on the extent to which the government enforces this law. If they start censoring games and cutting out huge chunks, then yes, that is a violation of the First Amendment. If they unjustly ban a game, that's a violation of the First Amendment. If, however, they deem a game violent and prohibit its sale to minors, that doesn't really violate freedom of speech.
The reason I'm opposed to this law is that legally enforced age ratings are completely pointless. A perfectly mentally healthy 12-year-old can't buy and play GTA, but a psychotic thirty-year-old convicted assaulter who's been granted probation can legally purchase and play it? Age really doesn't impact the ability of a person to handle violence/language/whatever.
But in Europe, it's not that bad, because at least there is a self-regulatory association which, despite erring on the side of caution a little, is mostly correct in judging the amount of inappropriate content in a game and uses a full range of age ratings. What is this law proposing to do, exactly? Set up a commission to decide which games are violent and which aren't? Or is it still the ESRB making the decision?