In season 5, episode 8 (Hardhome), we saw the Night King
seemingly resurrect the dead by the wave of a hand, but he stares at Jon so there is no line of sight.
So it is a fact that there is no line of sight necessary. In season 7, episode 6, we saw
the Night King seemingly reanimating a dragon by the touch of his hand.
One theory might be that the ability to resurrect diminishes with distance.
Also, I don’t think the white walkers spread a "disease" and don’t have to do anything to resurrect because otherwise, at the beginning of season 6,
Jon would have come back as a wight.
As far as resurrection distance goes, they probably can’t resurrect across the wall. If we assume the Night King’s motive is the destruction of humanity, it would have been over very soon.
At this stage, I have to admit that the motivation of the Night King is unclear (at least to me) - just having the ability is a different thing from actually doing it.
So, are there any other indications in the TV series as to how far their ability to reanimate the dead reaches?