Early on (for several hundred years), ancient Greek was written right to left, like Phoenician. Eventually, for reasons that are not entirely understood, it switched to left to right as you say.

But there was a transitional period which gives clues. Starting around the 6th century BCE, Greek was often written in alternating lines. So for example, the first line of an inscription would run left to right, then the next would run right to left, then snake back to left to right. Even the letters would be reversed in the alternating lines, much to the confusion of readers. An example in English looks like this. This script is called 'Boustrophedon' for 'as the ox turns,' because it resembles the furrowing of a field. Such a system was somewhat helpful to experienced scribes, who could continue writing on alternating lines without having to readjust.

During the Classical period, Boustrophedon fell out of style and left to right become the standard. Perhaps left to right was chosen arbitrarily, most likely by some informal standardization process (as neither perspective held supremacy in the Boustrophedon system).