In his fascinating book The Discarded Image, C.S. Lewis (yes, the same medievalist who also wrote the Chronicles of Narnia) talks about the worldview of the middle ages and how so many of the concepts that comprised it, the celestial sphere, hell and purgatory, the role of angels, saints and relics, derive in large part from the ideas of Greek and Roman philosophers, mediated by Roman neo-platonic authors of the late Roman empire and philosophers such as Boethius from late antiquity. When it comes to many of the key elements of medieval Christianity, where it differs from the early days of pre-Constantinian Christianity, they are heavily borrowed from the ideas of imperial Rome.
This can be seen as well in the art of the middle ages. The iconography of several depictions of Christ in medieval art are very reminescent of the art of the Roman empire. The iconic image of Christ enthroned surrounded by the twelve apostles is visually very similar to images of the emperor surrounded by his court that you can find on the Arch of Constantine in Rome or on the base of the obelisk of Theodosius in Constantinople. Late Roman depiction of Aristotle with his disciples becomes the iconography of Christ delivering the scrolls of the law to saints and martyrs. The equestrian statue of Marcus Aurelius is used as a model to depict the old testament Abraham in the mosaics of Santa Maria Maggiore in Rome. Over and over we see Christian art of the middle ages that borrows heavily from Rome, in part because in the beginning Christianity lacked its own artistic tradition, and in part because the visual culture of Rome was highly praised and became an important connection to the early centuries of Christianity. Borrowing the imperial iconography of power also helps depict the power of Christ over the heavens and the Christian world.
We see the same influence of imperial Rome in Christian architecture. Early Christianity did not have a tradition of identifiable christian worship buildings, and borrows heavily from the Roman world to the point that the Roman basilica becomes the blueprint for countless Christian churches in the middle ages and beyond.
The legacy of Rome at the end of the western empire is an intrinsic element of the medieval church, to the point that it is not far fetched to identify the Christian church as the continuation of the Roman world in the middle ages.