The crusades, originally intended to save the Byzantine Empire, among other things, failed to do this and hastened its decline instead. The crusaders sacked Byzantine cities and eventually captured Constantinople in 1204. They then used Constantinople as the capital of what is called the Latin Empire, which it remained, until the Byzantines managed to recapture it in 1264. The Byzantine Empire never really recovered from this, though it held on for almost another 200 years.
The underlying premise of the question is incorrect. The Byzantine Empire never converted to Islam and Islam was, by and large, illegal within the Byzantine Empire for as long as it existed. The reason why the territory formerly controlled by the Byzantine Empire became Muslim-majority was because it was conquered by various Islamic States such as the Rashidun Caliphate, the Umayyad Caliphate, the Seljuq Turkish Empire, and the Ottoman Empire (which finished the job).
Islam
Islam became a rival of Christianity in the Byzantine Empire.
The Byzantine Empire declined after Islam spread. The words "Islam spread" are nebulous. If the question is referring to the Rise of Islam period, when Mohammed and the Rightly-Guided Caliphs were spreading Islam, then yes, the Byzantine Empire decline after these events. If, however, you are referring to the Seljuk and Ottoman conquests of Anatolia nearly 600 years later which eventually ended the Byzantine Empire, the Empire was in decline long before these armies trampled the Byzantines to the ground. The answer is more complex than the above answers would purport. See the link below for another relevant answer.
The Seljuks or the Sufi's
Islam
The underlying premise of the question is incorrect. The Byzantine Empire never converted to Islam and Islam was, by and large, illegal within the Byzantine Empire for as long as it existed. The reason why the territory formerly controlled by the Byzantine Empire became Muslim-majority was because it was conquered by various Islamic States such as the Rashidun Caliphate, the Umayyad Caliphate, the Seljuq Turkish Empire, and the Ottoman Empire (which finished the job).
400 - 1400, during the era when the Roman Empire declined, the Byzantine Empire flourished, and Islam was founded.
Islam
Islam became a rival of Christianity in the Byzantine Empire.
The Byzantine Empire declined after Islam spread. The words "Islam spread" are nebulous. If the question is referring to the Rise of Islam period, when Mohammed and the Rightly-Guided Caliphs were spreading Islam, then yes, the Byzantine Empire decline after these events. If, however, you are referring to the Seljuk and Ottoman conquests of Anatolia nearly 600 years later which eventually ended the Byzantine Empire, the Empire was in decline long before these armies trampled the Byzantines to the ground. The answer is more complex than the above answers would purport. See the link below for another relevant answer.
Islam became the strongest rival of Christianity in the Byzantine Empire.
Invasions by the Islamic Ottoman Empire destroyed the Byzantine Empire.
The Byzantine Empire, as well as the Persian Empire were located near the Arabian Peninsula, in the budding years of Islam.
Invasions by the Islamic Ottoman Empire destroyed the Byzantine Empire.
The Seljuks or the Sufi's
Invasions by the Islamic Ottoman Empire destroyed the Byzantine Empire.