Of course. The Carolingian kingdom even used roman law, and as a direct result so did Charlemagne's 3 successor states and thus so did medieval France and the Holy Roman Empire.
And it's not like there wasn't a roman empire during the medieval period. While western Rome collapsed in the 5th century, eastern Rome did not, and the indigenous inhabitants of what we now in English call the Byzantine Empire considered themselves Romans and were considered as such by their contemporaries. When Odoacer took the crown of Rome he himself didn't claim the title of imperitor, he sent the crown to the eastern Emperor as he felt it was his rightful property. And furthermore hundreds of years after that both the Arabs and Turks called eastern Rome "Rum" and the inhabitants of the eastern Roman empire "Romans".