The papacy certainly saw (and arguably continues to see) itself as the continuation of the Roman Empire. They even had a (forged) document to prove it, the Donation of Constantine, which effectively said as much. The Byzantine Empire, which also (and probably correctly) also viewed itself as the continuation of the Roman Empire, would dispute that claim.

As for other institutions, it depends on how you define them. Almost all modern western law has its basis in Roman codifications, for example. Some parts of our modern education system have their origins in the classical schools.