Was homosexuality rarer in ancient Sparta than in any other Greek state?

Short Answer

There is no evidence of any law banning h*m*sexual relationships where one partner submits to the other, but Xenophon does mention the forbidding of physical pederastic relationships by the semi-mythical law-giver Lycurgus. The evidence that we have, though, indicates that Xenophon was stating an ideal rather than reality.

The supposed law 'no man shall submit to another' is more a reflection of the Spartan ideal that death was preferable to surrender on the battlefield, i.e. no man should submit in battle.

Details

While much has been written on the topic of h*m*sexuality, both in Sparta and in other Greek city states, the evidence is often second-hand, open to interpretation and / or quite possibly a reflection of an (ancient) author's bias. There is also much controversy among modern scholars (see Wikipedia for a summary and some examples).

Xenophon makes quite a number of references to h*m*sexuality in Greece, contrasting a Spartan man's appreciation for a boy's character with other Greeks' greater tendency to physical relationships. However, we have to keep in mind that Xenophon was an Athenian exile who was generally pro-Spartan, and his own sons went through the Spartan Agoge.

Keeping the above in mind, Xenophon says that the Spartan (semi-mythical) law-giver Lycurgus did not approve of physical h*m*sexual relationships between the erastēs ('lover', the older partner usually aged 20 to 30) and the erōmenos ('beloved', the younger partner usually aged 12 to 18). To quote Xenophon's Polity of the Lacedaemonians,

...if...it was not an attachment to the soul, but a yearning merely towards the body, he [Lycurgus] stamped this thing as foul and horrible; and with this result, to the credit of Lycurgus be it said, that in Lacedaemon the relationship of lover and beloved is like that of parent and child or brother and brother where carnal appetite is in abeyance.

D.M.MacDowell, in 'Spartan Law' (1986) argues that

there was no law banning h*m*sexual acts with boys in Sparta. Nevertheless, we can accept that moral relationships between boys and lovers was of greater significance and interest than physical ones,....

MacDowell also examins the evidence from Plutarch, and says that this seems more anecdotal than evidence of any laws, adding

the preferable conclusion is that the Spartans had no law on the subject of h*m*sexual relationships

McDowell's view is broadly in line with most other academic opinion; see Brian Bertosa, 'Sacrifice to Eros and Homosexuality in the Spartan Army' (2009) - the author also cites other academics in this article. Note also that Xenophon cites Lycurgus only on pederastic h*m*sexual relationships, not on relationships between adult men.

However, men were expected to marry and have children who would grow up to become either (for men) soldiers of the state or (for women) mothers of more soldiers, so it seems unlikely that h*m*sexuality was encouraged to the extent that it was with the Theban Sacred Band. In Sparta, failure to marry and at least attempt to produce sons bore heavy social penalties when a man passed thirty. To wit,

Under the law bachelors suffered a diminution of full civic rights and a fine, together with public disgrace and ridicule. They were excluded from the Gymnopaidiai festival (Plut. Lyk. 15.2) and so, I assume, from the holding of offices connected with its celebration. On public occasions not only would younger men not rise to offer them their seats, but the bachelors were obliged to surrender theirs to their juniors, a terrible humiliation in gerontolatrous Sparta (Xen. Lak. Pol. 9.5; Plut. Lyk. 15.3; Mor. 227EF; cf. Hdt. 2.80.1).

Source: Paul Cartledge, 'Sparta and Lakonia: A Regional History, 1300-362 BC' (2002).

Nonetheless, close non-sexual relationships between soldiers for military reasons were encouraged (and this almost certainly went beyond comradeship at times).

Paul Cartledge also states that the Spartans

... did not officially encourage, let alone institutionalize, for military purposes h*m*sexual realtionships between two men of adult fighting age.

Nonetheless,

The Spartans had a reputation among other Greeks, no doubt exaggerated, for being addicted to buggery.

This is perhaps typified by Aristophanes' play Lysistrata where the author makes several crude references to Spartan sexual preferences. However, Aristophanes was an Athenian playwright writing during the Peloponnesian War, and he was writing for comic effect. Thus, we cannot infer from this that Spartans practiced h*m*sexuality either more or less than other Greek states:

Five jokes about Spartan h*m*sexuality [in Lysistrata], then. But we must put them in perspective. There are dozens of jokes about Athenian h*m*sexuals, and indeed about Athenian sexual practices of all imaginable and unimaginable kinds: enough to fill a book, as indeed they have done.

Source: David Harvey, 'Lacomica: Aristophanes and the Spartans'. In Anton Powell and Stephen Hodkinson (eds), 'The Shadow of Sparta' (1994)