Writer Soo-Mahn wrote an article a little while back, that was much lauded and linked to about this topic. I loved the article, that cited Earthbound, Ico and Mario as good examples of games as art. He very effectively explained the transition a story has to make to become a compelling game, by giving players choices.
I want to forge my own path on this road. Here's my thesis:
Video Games, like all art forms, take a person with keen sense of vision, emotion, evocation and imagination to translate the written word or the formless thought into an enriching experience both engaging to the mind, and compelling.
Games have Designers for a reason. Designers are supposed to control and manipulate the game before the players eyes. They look at the situations in the game (any genre) from the eyes of the artist, and the eyes of the viewer. While some artists merely make art for themselves, a lot of artists take their audience into account when creating. They want to sell paintings. Designers are the same. They need to craft an experience that will hold people's attentions, draw them in and amuse their whimsy for at least a while. Whether the game is
Shadow of the Colossus or
Grand Theft Auto IV, two games from two entirely different ballparks. The Designer holds the brush, and with it they try their best to paint a picture worth seeing (More on Designers and sacrifice later...). I want to talk about the picture itself for a little.
The game as an artistic creation must be looked at from a few angles. The above article points out that a good game will typically offer the player choice. Instead of a movie/play/book where A always leads to B, in a game A could lead past B to C. The player will decide what goes. I think this is a great point, but there is so much more to a game as art. It can't just be left at that. Leaving it at that makes Choose-Your-Own-Adventure books art. Which some really are! But I digress. The look of a game, past it's actual game play flow, is essential to defining a game as an art form.

Take for example the Clover Studios creation
Okami. Easily one of my top 5 video games. Beautifully made, amazing to play. Compelling story, sights, sounds...it's really a cream of the crop title. Is it art? Indubitably. The game from start to finish has the mark of the craftsman on it. The mark of an individual who knows the world through the eyes of the gamer, and knows how a game needs to look, as well as play. Every scene from
Okami has care in its design. There isn't a single moment of that game that doesn't look like it could be taken aside to be a painting. The same goes for games like
Shadow of the Colossus and
Ninja Gaiden II in their own style. The hand of the artist is all over those games. The hand of the artist is imperative to anything being truly called art.

If I had designed
Ninja Gaiden II, it would have been different in fundamental ways. Because I did not, and Itagaki did, it is an Itagaki. If I re-painted
A Sunday Afternoon on the island of La Grand Jatte it would have been different, and not just because I'm bad at pointillism. In that sense, every game is an original work of art by the people who make it, who hopefully put care into people's perception of it.
Art is based off of experience, linear or not. The experience of seeing a beautiful painting by
Jack Vettriano mimics the experience I get playing
Okami when I've succeded in making the trees blossom, or in
Grand Theft Auto IV when I witness Niko trying his damndest to succeed in a world ready to make him become the devil himself. Video Games give you the experience of experiencing, which may not make any damn sense at all. But if it does, I'd like to pretend it means that merely because you are there, making choices and participating, it'd be like walking among those people on La Grand Jette, talking and smiling and playing just as I do in my imagination every time I see it in person.
I hope more designers realize this, and take this into account. Games need to be an experience for the gamer, in a good way. Don't hold their hand, don't coddle them...but give them something that they can take away, and something to draw them back. All good art leaves a mark on the soul, and the medium of games should do no less.