The evolution of the Latin Rite as we know it today does not necessarily come from Sacred Scripture as much as it does ecclesiastical functionality.
As The Roman Rite aritcle at The Catholic Encyclopedia puts it:
The Roman Rite was adopted throughout the West because the local bishops, sometimes kings or emperors, felt that they could not do better than use the rite of the chief bishop of all, at Rome. And this imitation of Roman liturgical practice brought about in the West the application of the principle (long admitted in the East) that rite should follow patriarchate.
Apart from his universal primacy, the pope had always been unquestioned Patriarch of the West. It was then the right and normal thing that the West should use his liturgy.
The Church does, as it always has, teach that the the Greek language used by the New Testament writers is the inspired "language" chosen by God to reveal his Truth. Prior to the invention of the printing press only about 10% of the population could read. This number dropped even lower with the advent of the "dark" ages initiated by the Barbarian invasions and the fall of Rome.
The New Testament as we know it was not officially canonized until A.D. 397 with the ratifying approval of Pope Damasus. By this date the Mass had already been celebrated in and around Rome for over 300+ years prior. St. Jerome was commissioned by Pope Damasus to translate the newly canonized Greek texts into Latin so that more people could understand the New Testament texts. This is the same reason why the King James/Douay-Rheims editions have been translated into the more universally spoken English language of today.
(See Canon of the New Testament from The Catholic Encyclopedia.)
Keep in mind that the Eastern Catholic (and also Greek Orthodox) churches have all along used Greek in all of their liturgies and rituals to this day. It is arguable that the language chasm that existed between the East and West is what ultimately lead to the East-West Schism.
Later, during the Reformation period, the Council of Trent was convened to address and severely condemn the rampant abuses that had crept into the Church's liturgies, as well as respond to the Reformers. The Council led to the solidification of the Tridentine Mass, which was celebrated universally in the Western Latin Rite for over 400 years until Vatican II. Elements of the Tridentine Mass still exist in the "Novus Ordo" liturgy, which uses both Latin and the vernacular of the local region (i.e. English, German, French, or Spanish).
(See Reforms of the Council of Trent.)
Latin's superiority and precision is 2nd only to Greek in the history of languages. Perhaps this is why the Holy Spirit chose Greek for divine revelation. This is also why the Catholic Church was hesitant to translate the Bible into English from Latin. English is more likely to be manipulated to misrepresent meanings conveyed. Latin is more pure than English. Greek, however, is the purest of all languages.