I'm a keen darter, so an interesting post (note I said 'keen', not good!) Compare it to shooting or archery. You are hitting, or aiming at, a target from a fixed distance away from it. Is shooting or archery described as a game? It is played in olympic competitions so probably it gets the full 'sport' treatment.
Darts has come on a hell of a long way since the days of Jocky Wilson and Eric Bristow slogging it out in smoke filled clubs, with pints of lager as refreshment. The likes of Phil Taylor train in gyms just to get themselves fit enough to play on a stage in baking heat for a couple of hours. This has served well to raise the profile of the players.
The formation of the PDC has resulted in the darts players who choose to go 'pro' being in a better position than ever before with hundreds of thousands of pounds of prize money available on the circuit. This, in my opinion, makes this particular side of it a 'sport'.
But I suppose I wouldn't call having a few games of 501 with your mate over a few pints in the local a sport, it's more of a game then.
Now, let's cover some more "pub olympics" - what about pool and snooker?