Question: I was just reflecting on the fact that Spanish, Portuguese, Italian, and French all evolved from Latin, the language spread across that area by the Roman Empire 2000 years ago. (Romanian too, I suppose.)

Yet although Greece was firmly a part of the Roman Empire, Latin never took root there, it seems, as the Greek language is quite distinct from the Romance languages. The same is true for the territories above Greece-- that whole region east of the Adriatic aside from Romania.

Why is this so?

Answer: 
Suspect it was because Greek already had well established literacy whereas other regions you name were much less literate. The Latin alphabet was based on the Etruscan alphabet which was in turn based on the Greek alphabet. 

If you were going to rule the Greeks you were going to have to do it in Greek.

To those areas you name Romanization brought both the Latin language and the alphabet.

To the extent that Spain for example was already literate was due to colonies of other nations... there were Greek and Carthaginian colonial cities/trading centers in Spain where they wrote in the the tongues of their colonizers.
The old Greek empire, meaning the empire of the Macedonians, the empire created by Alexander spoke Greek widely and had many Greek colonies. The entire city of Alexandria, which was founded by Alexander and became the center of learning in the Western world for 500 years was entirely composed of Greeks and Greek-speaking people. The political control of Rome did nothing to change this.