So two years after being coronated by the Pope in 800, Charlemagne sent a marriage proposal to Irene, who was the mother of Constantine VI and de-facto ruler of the Byzantine empire. The fracture between Eastern and Western churches had been supposedly healed at the seventh Eucumenical Council held at Nicaea, but Pope Leo III was not satisfied and put his faith in Charlemagne. Part of why they viewed the Throne if Emperors as open is that Irene, as a woman, was de barred from holding it per Salic tradition. It also helped that Irene was a renowned murderess.

Why did they choose to do this? Well, Pope Leo was charged with multiple serious crimes, such as perjury and adultery, and the only person considered worthy to judge the Vicar of Christ might was an Emperor, and it definitely was not going to be a woman. And so he crowned Charles the Great, and now the Pope had the authority to appoint the Emperor of the Romans.

I'm just going to directly quote the response: "The Byzantines, on the other hand, saw things rather differently. To them, Charle's coronation was an act not only of breathtaking arrogance, but of sacrilege. Their Empire was built on a dual foundation: on one hand, the Roman power; on the other, the Christian faith. Thus, just as there was only one God in heaven, so there could be but one supreme ruler on Earth."

Upon finding out that Irene was seriously considering marrying Charlemagne, high ranking officials deposed Irene and exiled her to Lesbos. They did not view Charlemagne or his coronation particularly kindly.

Source: John Julia Norwich "A Short History of Byzantium"