The Trojan War was not a war between Sparta and Troy, even if Helen was the wife of King Menelaus of Sparta. The traditional Greek version of the war, for what it’s worth, has a coalition of warriors from all over Greece taking part, led by Agamemnon, king of Mycenae and brother of Menelaus. The most memorable Greek heroes were not Spartans—men such as Achilles, Odysseus, Nestor, Ajax, Diomedes.
There is abundant evidence to indicate that Troy, and the Trojan war, didn't happen in the Mediterranean at all, but in northern Europe, perhaps as a struggle of early Celts; and that Homer was merely reciting his version of an older conflict, the news of which has passed widely throughout ancient Europe. I suggest a reading of the book Where Troy Once Stood, by Iman Jacob Wilkens, for a good exposition of an alternate interpretation of the particulars of that conflict.