What does being Indian mean to Victor?

Victor has an uneasy relationship to his Native American roots and heritage. On the one hand, he knows that he must leave the reservation behind and head on out into the wider world if he's to lead anything like a fulfilling life. But on the other hand, this initially makes him feel disconnected from his roots, lonely and unsure of himself in white society.
More than anything else, for Victor being an Indian is about feeling that you never really belong anywhere. That doesn't just apply to Spokane and Seattle, but also to the reservation on which Victor grew up. The reservation, like all such places, is a completely artificial community, one that separates Native Americans from the land of which they've long since been dispossessed. It is only by combining his many painful experiences of life on the reservation, with his no less painful experience of living in white society, that Victor's finally able to feel a sense of wholeness and inner peace and return to the reservation with a sense of direction in life.

In the short story "The Lone Ranger and Tonto Fistfight in Heaven," the unnamed Native American narrator travels through late-night Seattle while dealing with a recently ended relationship with his white girlfriend. Throughout his journeys, it is clear the narrator is dealing with a kind of guilt based on this relationship and the separation from his roots on the reservation. He journeys throughout the city, sometimes getting lost, saying, "Seems like I'd spent my whole life" looking for something familiar.
The narrator eventually reveals the fact that since he is Indian, he doesn't belong in Seattle with his white girlfriend:

There's an old Indian poet who said that Indians can reside in the city, but they can never live there. That's as close to truth as any of us can get.

This guilt and lack of belonging leads the narrator back home to his Spokane reservation. The narrator returns to his roots—including basketball—and finds a sense of himself, eventually accepting his limitations and then journeying back out to the white world.
Sherman Alexie often uses salmon, which served as a source of life for his tribe, as a talisman for his characters' lost roots, and this is no different here. At the end of the story, the narrator says he is back in Spokane but wishes he "lived closer to the river, to the falls where ghosts of salmon jump."