I have never seen a compelling argument for Sparta embracing homosexuality. but Paul Cartledge and some other renowned Sparta scholars claims there was and they certainly know more about ancient Sparta then I do.

Sparta as elsewhere in Greece had an institution were boys formed a relationship with a adult man. The purpose of this bond was primarly pedagogical, the man should guide the boy to adulthood, yet in many places and circumstances it had a sexual dimension as well. The general greek term for this was erastes-eromenos, lover-beloved. the Spartan used the terms eispnelas, Inspirer, aites, Hearer which might give modern people a more reasonable picture of it.

The two main sources we have for Spartan culture, Xenophon and Plutarch both categorically deny a sexual aspect in this relationship and that if the mentor would use his charge sexually he would be killed/exiled/lose his citizenship. according to Xenophon, Lycurgus viewed such a relation comparable to incest.

I think I ought to say something also about intimacy with boys, since this matter also has a bearing on education. In other Greek states, for instance among the Boeotians, man and boy live together, like married people; elsewhere, among the Eleians, for example, consent is won by means of favours. Some, on the other hand, entirely forbid suitors to talk with boys. The customs instituted by Lycurgus were opposed to all of these. If someone, being himself an honest man, admired a boy's soul and tried to make of him an ideal friend without reproach and to associate with him, he approved, and believed in the excellence of this kind of training. But if it was clear that the attraction lay in the boy's outward beauty, he banned the connexion as an abomination; and thus he caused lovers to abstain from boys no less than parents abstain from sexual intercourse with their children and brothers and sisters with each other.

Xenophon, Constitution of the Lacedaimonians, chapter 2

Plutarch when addressing the bond says this :

Affectionate regard for boys of good character was permissible, but embracing them was held to be disgraceful, on the ground that the affection was for the body and not for the mind. Any man against whom complaint was made of any disgraceful embracing was deprived of all civic rights for life.

Plutarch, Customs of the Spartans, 7

Now Xenophons account is particular hard to dismiss because he is the only source which had undisputed extensive access to lycurgian spartan society, he spent many years living there. even led Spartan troops in battle and sent both his sons through the agoge, unlike all other sources he cannot be mistaken when he says Spartans abhorred pederasty, only lie.

Xenophon was a spartan fanboy, he left Athens because he admired them so much and his accounts distinctly bends the truth in their favor(might keep in mind that few if any ancient sources lack heavy bias). Cartledge and other seems to simply assume he is covering up all the boysex to paint a better picture.

The problem is why would he? If Sparta really find homosexuality something positive why isn't he praising it as he does all other aspect of Spartan culture? "abomination" "comparable to incest" You think all those literate spartan kings and aristocrats who Xenophon are dependent on, would be seriously offended by such a castigation of their wholesome institutional homosexual practices.

That he tries to hide it from the rest of Greece makes no sense either when he continues I am not surprised, however, that people refuse to believe this. For in many states the laws are not opposed to the indulgence of these appetites. As in pederasty is normal in ancient Greece so hence why would the Spartan practices be so shameful that they had to be hidden?

Nor can he have considered this supposed Spartan pederasty very troublesome since he sent both his son's through the Agoge to partake in this sexual abuse. At the very least the easiest way out of it for him would have been to say nothing about it.

Then we have Aristotle again a contemporary source who with great knowledge about most things and who who has a lot to say about Sparta in it's constitution. Mostly about what the Spartans are doing wrong. In one segment he explains the deplorable influence and power Spartan women have over the men with this :

And this is what has actually happened at Sparta; the legislator wanted to make the whole state hardy and temperate, and he has carried out his intention in the case of the men, but he has neglected the women, who live in every sort of intemperance and luxury. The consequence is that in such a state wealth is too highly valued, especially if the citizen fall under the dominion of their wives, after the manner of most warlike races, except the Celts and a few others who openly approve of male loves. The old mythologer would seem to have been right in uniting Ares and Aphrodite, for all warlike races are prone to the love either of men or of women. This was exemplified among the Spartans in the days of their greatness; many things were managed by their women. *

Aristotle, Politics, book 2 chapter 9

That's to say, the Spartan mens exceptional libido and heterosexual preference is the reason the Spartan women has so much power in Sparta. Now you might find Aristotle's explanation of Spartan gender dynamics strange, but he or his reading public clearly didn't believe Sparta approved of "male love".

Cartledge response to this in his book "reflections" is something like "yeah but the Spartan disapproved of homosexuality between adults but it says nothing about sex with boys."

Yet the core of Aristotle's argument is that Spartan men lived under exceptional sexual constraint with female empowerment as a consequence, should boys have been a sexual outlet in any meaningful way his explanation for Spartan female domination relative other Greek societies makes no sense.

Nor do I think encouraged socially acceptable homosexuality made much sense from what we know about the rest of Spartan society. Ancient Greeks didn't have more gay genes, they practiced pederasty for the same reason you see it in Afghanistan today, women were kept away from the public and covered in the rare cases they left the house. Greek men didn't marry until their 30thies when they amassed enough wealth to support a family. This presumable left young men with some considerable sexual frustration.

In Sparta this was supposedly very different:

..he ordered the maidens to exercise themselves with wrestling, running, throwing, the quoit, and casting the dart, to the end that the fruit they conceived might, in strong and healthy bodies, take firmer root and find better growth, and withal that they, with this greater vigour, might be the more able to undergo the pains of child-bearing. and to the end he might take away their over-great tenderness and fear of exposure to the air, and all acquired womanishness, he ordered that the young women should go naked in the processions, as well as the young men, and dance, too, in that condition, at certain solemn feasts, singing certain songs, whilst the young men stood around, seeing and hearing them.

..Nor was there any thing shameful in this nakedness of the young women; modesty attended them, and all wantonness was excluded. It taught them simplicity and a care for good health, and gave them some taste of higher feelings, admitted as they thus were to the field of noble action and glory.

..These public processions of the maidens, and their appearing naked in their exercises and dancings, were incitements to marriage, operating upon the young with the rigor and certainty, as Plato says, of love, if not of mathematics.

Plutarch, Life of Lycurgus

Both Xenophon Plutarch and Aristotle have mutual accounts about how Sparta strongly enforced sex for procreative purposes. Men with heirs, sires, didn't rise for older men without. bachelors were excluded from the great festives of gymneopedia and presumably for the state offices awarded there and forced in winter time to march around the city naked mocked by the women.

Aristotle tells us that the Spartan's had a law that excused a Spartan with 3 sons from military service and 4 from all taxes as well. Despite this the numbers of Spartiate's continued to decline. So while they where punished for lack of procreative sex and rewarded for keeping their wives pregnant they would also have been required by the state to have sex with boys for teambuilding purposes? Must have been rather confusing all-round.

Might also be worth nothing that while you find homosexual depictions in art from Corinth, Thebes and Attica, none has been found or made in Sparta.

Follow-up: were spartan homosexual relationships (if they were as common as sometimes portrayed) almost always pederastic, or was it normal for adult men to have sexual relationships too?

Because Xenophon was a primary source, shouldn't he have more credence[?]

There are all sorts of reasons why scholars don't uncritically believe primary sources, especially primary sources from the classical world.

The problems with our sources on Sparta in particular are so great that there is a historiographical concept specifically dealing with it, known as the Spartan mirage. Basically, the problem is that our sources are all non-Spartan (well, in the archaic period we have Tyrtaeus, who was a Spartan poet; and we have some archaeology, but other than that...), many of them were writing about what was to them the past, and they frequently had an exoticising and idealising agenda, and focus on aspects of Spartan life which seemed unusual. Xenophon is one of our better sources in the sense that he was in a position to know what was actually going on in Sparta in the classical period, but he still had a strong laconophile ideology and reason to portray the Spartans idealistically.

And, while homosexual pederastic relationships were acceptable in classical Greece, there was still an ideology, at least in Athens, where Xenophon was originally from, that these relationships should not actually be sexual (which, as /u/XenophonTheAthenian points out elsewhere in this thread, is what Xenophon denied).

I don't know enough about the history of Spartan pederasty to have a strong opinion on whether or not they had sexual relationships, but this might give you some idea of why historians don't necessarily uncritically accept Xenophon here.