It is accepted history that the Romance languages spoken in various European countries are descended from Vulgar Latin and are the result of Roman conquest and colonization of those areas. Yet it is widely believed that the Koine Greek was commonly spoken among the Roman legions. It may have been that after the battles of conquest, the legions withdrew and Italians were imported to take over the administration of what they had just conquered. So that's why Spain is not a Greek-speaking country. Or maybe the Greek-speaking legions were mainly in the Middle East. So in Spain and other Roman colonies the subjects eagerly adopted the language of their conquerers. Was Basque the language of the Iberian peninsula before the Roman conquest? I would guess that Phoenician was also spoken in some parts of pre-Roman Iberia. Your guess is as good as mine. In Dacia the pre-Roman language was Slavic. This could be why Rumanian has some Slavic words in its vocabulary.

Good question. I often wonder why the Ibero-Romance languages have 2 phonemic rhotics, a trill & a tap, when the Romans apparently confounded their single rhotic with their lateral. I've noticed that Basque also has the distinction between 2 rhotics. Could the 2 rhotics of Ibero-Romance have come from Basque influence?

Other features I've heard ascribed to Basque influence are the five-vowel system of Spanish and the shift of initial f > h > ∅.

Wait, Latin had more than 5 vowels?
(bear with me, guys, I know next to nothing about Latin)

Classical Latin had the same vowels as Spanish /a e i o u/ with a length contrast, as well as a few diphthongs. Vulgar Latin however had a contrast between open-mid and close-mid vowels which is still maintained in French, Portuguese and Italian but has been lost in Spanish (and Romanian).

I didn't know Italian also had them.

Is it possible for the double rhotic to have evolved from the contrast of consonant germination?

That sounds plausible.

As far as Basque is concerned, it is generally understood to be the living descendent of Aquitanian, the pre-Roman language of Aquitania or Vasconia (the area between the Garona/Garonne and the Pyrenees), from which comes the Occitan name Gasconha and its French and English adaptations Gascogne and Gascony. The indigenous language of the Gascons, likely descendants of the ancient Vascones, shows similarities in its phonological evolution to the Spanish dialect of Castile, which through geopolitical chance was imposed on most of the rest of the Iberian peninsula. For /f/ in other Occitan regional varieties, Gascon almost exclusively has /h/: "Lo men hrair que vedó lo son hilh a la frinèsta/hrinesta/hièstra" instead of "Lo meu fraire vegèt son filh a la fenèstra" (My brother saw his son at the window").

There seem to be links between Basque (via Aquitanian) and Iberian, but these could just as well be due to borrowing as to a genetic relationship given the present lack of knowledge of the Iberian language. (Look up "Iberian language" in WIkipedia for a primer.)

There were a couple of other pre-Roman languages spoken in the peninsula: Tartessian and Celtiberian (as far as is known nowadays).

As for the reasons why Latin and then Romance (Vulgar Latin) took root instead of Greek, I think the fact that Latin was THE single official language of the empire probably had a big role to play. Don't forget that Emperor Cicero was from Tarracona (now Tarragona). Several other emperors were from outside Italy as well.