The island kingdom founded by the Men of Westernesse in Tolkien’s legendarium is Númenor.
Númenor, also called Westernesse, is an island realm in the Second Age of Tolkien’s history. The Valar raise it from the sea as a reward for the Edain, the Men who aided the Elves during the wars against Morgoth. Elros, the half-elven brother of Elrond, becomes its first king.
The Númenóreans are granted unusually long lives and great skill. Over time, however, their fear of death becomes increasingly powerful. Their rulers grow resentful of the immortal Elves and eventually fall under the influence of Sauron, who is taken to the island as a prisoner but becomes a counselor to its king.
Númenor is destroyed in the Second Age when Ar-Pharazôn attempts to challenge the divine order. Survivors establish the kingdoms of Gondor and Arnor in Middle-earth.
Númenor is therefore the ancestral source of the Dúnedain, including Aragorn, rather than simply another name for Gondor.