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It doesn't matter how you call it. If it makes enough money, people we'll want it to exist and may call it whatever they want. It's simple as that.

The definition of sport is "An activity involving physical exertion and skill in which an individual or team competes against another or others for entertainment." according to oxforddictionaries.com

So if that's the definition of sports you have, most games that have an esports scene could be counted as sports in general, yes.
It doesn't specify that the whole body needs to be exerted, and the fingers/hands/arms and brains definitely are, if you do it competetively.
Most such games also require very good reactions, hand-eye-coordination and the such.

I guess I won't have to go ino "competes against another or others for entertainment", anyone would see that that is given ^^