"Islands" is plural, meaning that there is more than one island.
"Island's" with the apostrophe before the s is possessive, meaning that something belongs to one island.
"Islands'" with the apostrophe at the very end, is possessive plural, meaning that something belongs to two or more islands.
An island can be a state, and a state can be an island. Like Hawaii.
But if you're going to make a name of a country, and the country is made up of islands, some possibilities for a pleasing sound might be "The United Islands of....." or "The United Island States of...." or even "The United States of the.....Islands". You just need a name to replace the "...".