I don't know how much is needed. Parts of Sweden were in the Baltic during Viking times but are now farmland, so perhaps no more than a few hundred years? Then again, there's more rain in those areas of northern America and Europe where the glaciers were than in the Mediterranean.
I found https://en.wikipedia.org/wiki/Salting_the_earth which suggests that the salting was more symbolic, or a curse, and that "There is no evidence that sufficient amounts of salt were used to render large tracts of land unusable."
Given how expensive salt was, it seems like it would be horrendously expensive to do that in the classical era.
Agricultural water runoff is a thing. The water doesn't all evaporate. Quoting http://www.watereducation.org/post/salinity-central-valley-c... , "Every time a farmer irrigates a crop, salt ends up concentrating in the runoff and groundwater."