While it's true that the island of Cyprus in recent times had had a majority Greek-speaking, Orthodox Christian population, it also has a significant Turkish-speaking, Muslim population as well. When Cyrprus was a British colony about a century ago, this latter group was something like a quarter of the population, and even after independence in 1960 it was about 17% of the population.

This community was significant enough in size that a serious political conflict developed between it and the Greek Cypriot community (the Turks favored some sort of federal or communal political structure that would give them a say in governance, while the Greeks favored majority rule). Major communal conflicts broke out in the 1960s, and a coup directed by Greece's then military junta seized control of the country in 1974, with the aim of unifying the island with Greece. The subsequent Turkish invasion led to mass population transfers and a partition of the island between the two communities.

Which perhaps doesn't exactly answer the original question. But perhaps Cyprus isn't so unique for large islands in the Aegean/Eastern Mediterranean. Crete also had a fairly significant Muslim population in the late 19th and early 20th centuries, but it suffered a fate different from the one of Cyprus - it was deported to Turkey as part of the massive, forced population transfers under the terms of the 1923 Treaty of Lausanne. Of course this is because Crete had been de facto annexed by Greece from the Ottoman Empire in 1908 (de jure in 1913), while Cyprus was annexed as a British colony in 1914 (but had been occupied by Britain since 1878)..