The ‘West’ created a seemingly-Greek culture, not Greek, by adopting and cultivating to the point of hybris only some elements of Greek thinking. Thanks to his deep study of the Greek language, Heidegger understood various differences between Greek and Western thinking, yet remaining himself a Western.
Rome is based not even in veritas, but in Lex (Law), which by itself reveals the foundations of the West to be conventional and subject to a permanent struggle between solipsism and objectivity (or between Descartes and Papal Infallibility, if you prefer such a scheme). All these are only marginal in Greek culture, interested (from Homer, to Plato, to the New Testament, to the Byzantine Fathers...) in the Person, the Truth as a Person (God, Demigod, Saint) and in Immortality as eternal Friendship with the Divine Person of God and Man.
Heidegger understands that Greek a-letheia is the uncovering (remembering) of a real life, yet without participating in this life, without even following Greek history, just studying some aspects of it, mainly linguistic, in a laboratory. The West created a culture that resembles a natural being, but beyond this resemblance, there prevail all those features which prove it precisely a monster.
If you know Greek, you can read two texts on these problems, the Ancient Greeks, and the Bio-Mechanical Intension. In the second text there is also an examination of modern European ‘philosophy’, including Heidegger, Husserl, Descartes, Spinoza, Schelling, Kant and others, as well as a presentation of the Papoprotestant approach to Christianity.