No it's not a sport, sport's already been defined.. it's rugby, football, etc.
Gaming popularity has gone mental now tho, particularly with Starcraft II. Millions of people around the world recognise the term "E-Sport", and so.. it's an e-sport. Plenty of players are full-time gamers with teams, sponsors, and fans. There's plenty of cash in it for the best of the best.
Doesn't mean you have to see the value in it, I don't see the value in football, but I spend several hours every single day spectating Starcraft II. If you want to get down to the nitty gritty of what it takes to be an e-sport, come back to me when you have 300 APM.
I regularly watch tournaments with tens of thousands of spectators, 50k is very often hit. To put that into perspective, 60k is the capacity of the Emirates Stadium.