The Short Version
Sicily's status as the largest island entirely within the EU is well-supported by geographic and political evidence. Every European island larger than Sicily (~25,700 km²)—Great Britain, Iceland, and the island of Ireland—falls outside the EU or is split between EU and non-EU jurisdictions. The counterargument that the Republic of Ireland should count conflates a political entity with a geographic island; the island of Ireland as a whole is not entirely EU territory due to Northern Ireland's UK status.