I decided to look up the definitions of sport and athlete. (Using define: "___" in google.)
Sport - an activity involving physical exertion and skill in which an individual or team competes against another or others for entertainment.
Athlete - a person who is proficient in sports and other forms of physical exercise.
Using these definitions, I'd say some video games, like those on Nintendo Wii, could be considered "sports" because physical exertion is present. The term athlete, when using the definition of sport, would allow the term to be applied to Wii players.
Without the dictionary terms, however, I wouldn't consider video games "sports" or their players "athletes."
I think the term being thrown around, "eSports" would more accurately define what we're talking about. As long as the "e" stands for "electronic" as implied, I could understand calling players "eAthletes."
Would I refer to these players as eAthletes in any sort of real life conversation or setting?
No. It sounds stupid.
I'd consider bowlers, pool (billiard) players, and even table tennis players as athletes because physical exertion and skill are required and understood to be necessary to play. While some games you can exert energy, its not a requirement.