Yes: Sparta was an Ancient Greek city, and the inhabitants of Sparta spoke a dialect of Greek and participated in the Olympic games. So Spartans were ancient Greek people, in the same way Persepolitans were ancient Iranian people. The more usual ancient designation of those people, though, was Laconians or Lacedaemonians, referring to the region that Sparta controlled. (Hence Laconic wit, referring to the stereotypical terseness of Spartans.)
Sparta was a prominent city-state in ancient Greece. In antiquity the city-state was known as Lacedaemon, while the name Sparta referred to its main settlement on the banks of the Eurotas River in Laconia, in south-eastern Peloponnese.