The problem with that is that this predictive power only manifests itself when you can make precision measurements of celestial objects, and the Greeks could not because they didn't have telescopes. (Hm, inventing the telescope might be an answer. Did the Greeks have lens-making technology?)
> Euclidean geometry as a starting point would blow their minds.
Sure, but that wouldn't prove you were from the future any more than it proved that Euclid was from the future.