Well, considering the various responses this discussion has received thus far, as better question to ask might be "what constitutes childishness?"
In the end, video games are a form of entertainment media. I'd put them in the same category as movies, tv, music, and books. That's not to say that all five are the same; they each have distinct differences that set them apart, and even that's not an apt-enough description.
Why should one form of media be considered childish then? Is said media to be defined by one such broad generalization?
Couldn't movies be considered childish in their own way? Of course that industry has been with us a LONG time but I don't think all movies are the same. Where does Citizen Kane fit in? Lord of the Rings? Schindler's List? Rambo? The Expendables?
What about TV? For every Breaking Bad & Game of Thrones you have Two & A Half Men and Family Guy.
Music? Of course I wouldn't say it's childish. What about genres? Where does classical fit in? How about pop or hip hop?
Books? Are we to say that the great literary pieces of history are to be compared to Curious George titles?
My point? Video gaming can't be defined by such a broad term. To say that it is childish is to cast some of the greatest titles as merely toys for children; in a sense. Is Grand Theft Auto childish? In a sense yeah, but it shouldn't be played by children for obvious reasons. How about other gritty games like The Witcher & Dragon Age? Yeah the fantasy violence can be seen as childish, but the childish brush stroke is a tad unfair in reference to characters and storyline.
But then, how about games like Journey? Shadow of the Colossus? Dark Souls? Ocarina of Time? Red Dead Redemption?
If you want a straight answer from me; you're not gonna get one. Can games be childish? Of course, and there's nothing wrong with that. My favorite game series of all time is Spyro the Dragon and for the most part it's downright childish. However, using that term for some of my other favorite games like Age of Empires or Dragon Age: Origins is a bit unfair. Yes, the case can be made, but I don't believe it's accurate. I don't play Witcher because I feel it's childish. It's fun of course, but the choices a game like that present to you is far from childish.
Let me reiterate: childishness in games is not a bad thing, it's just bad to label video gaming as childish, IMO.