Did citizens of the Byzantine Empire refer to themselves as Roman?

Yes, exclusively up until the late period (1204-1453). The Byzantines viewed themselves as the exact same people as the western and more ancient Romans. A common name applied to the Byzantines in Byzantine literature is "Ausones" or "Ausonians"^1, which comes from an ancient Greek term for people from Latium.

In the 13th century though, there was an unprecedented revival in ethnic Greek identity among educated Byzantines^2. This was a result of increasing anti-"latin" tensions since the disorderly conduct of the first crusade, which were strongly magnified after the crusader sacking of Constantinople. Byzantine scholarship was also going through a period of intense Hellenophilia, which played its part. Those Byzantines that adhered to this belief still had a Roman political and cultural identity, but believed themselves to be the descendants and heirs of the ancient Greeks^3.

It's important to note that this was a phenomenon largely confined to the educated upper classes in the Byzantine period. Isolated Greek speaking populations, such as that of the isle of Lemnos, continued to identify as Roman, as opposed to Greek, right up until the early 20th century^2.

^1 examples can be found in De Ceremoniis and the History of Attaleiates, among others

^2 see "Hellenism in Byzantium" by Anthony Kaldellis

^3 see Theodore Metochites' 13th century "Lament on the decline of Rome"