A proper way how to start this would be with a definition of art. However, art is a difficult thing to define. There are many opinions on this, some even hinting at the idea that the commonly understood concept of art does not make sense, that there is no meaningful distinction between art and craft (or art piece and product). But I actually don’t need an art definition to make my case about games being art.
No matter where you stand on this, we all recognize the traditional forms of art – painting, music, sculpting or architecture, literature, performance such as theater or dance and maybe even culinary. In the 20th century we saw the spread of movies, a combination of previous arts – painting, music, literature (i.e. storytelling) and performance, all glued together by a new art form we call cinematography.
And games are pretty much the same as movies. A combination of other art forms with new art form that makes it all possible – game design. While movies added motion, games added interactivity. Actually, if we won’t talk just about video games but games in general, board games and sport games predate movies. Just like other arts, games take skill to make and they evoke emotions in people. A well designed game can bring great feeling of satisfaction from beating the challenge and deeper enjoyment of story through interaction with its characters.
One thing to note - the interactivity (i.e. gameplay) is the most important part because that is what makes something a game. That is why I think Carmack was right about the importance of a story in a game. I see the appeal of playing a game for its story, because it is a different experience from movie or a book, but you still want the interactivity. If you only walk and collect pages of text, you might as well just read a book.