The problem with everything that we have about that period of Spartan history (which is often forgotten) is that it was all written from a Athenian perspective. And the Athenians at this period were at war with Sparta, and even if they were sympathetic they had most likely never even been - they were working from an Athenian image, for an Athenian audience. (Even in the multiple edited volumes by Steve Hodkinson this simple fact is often missed - though I have't read any works on Sparta written in the last 3 years, so I may be completely out of date on this now and am happy to be corrected)

Most of our information on Spartan social structures comes from Xenophon's constitution. But I don't really think it can be taken too seriously as a historic account. Look at it's circular structure (everything was great in the past now it's all terrible) which mirrors exactly his fictional education of Cyrus. And while Xenophon is taken more seriously now as a historian the hellenica (when it comes to describing Sparta) is still not that great. (this is also the main source Plutarch works from - Plutarch once described by Graham Shipley as a man with a big library but he didn't always know what to do with it).

Herodotus and Thucydides talk about Sparta as a militaristic society - but nowhere near as extreme as we like to think of them. Thuc.1.70 - the Spartans are described as passive, timid and cautious, also echoed in 1.84 (though this is in the context of a speech). Thuc.1.77 the Spartans are accused of not following their own laws, echoed in 1.95. My copy of Herodotus is in my office so I can find some relevant passages later (such as them loosing battles and the likes).

People also take fragments of Ephorus, but these have all the problems that fragments usually have (he's also accused of being a bad historian by Polybious- who is also the main reason that he survives).

They also use Plato's republic (the timocracy is taken to be a description of Sparta) and the laws (in which there is a Spartan interlocutor). Aristotle discusses it in his politics in much less glowing terms than Plato. His constitution doesn't survive. But these (Plato in particular) are descriptions of ideal societies - as a philosopher wanted them to be not how they really were.

Basically, this is a very round about way of saying - when it comes to Sparta, we just don't know. The whole notion of an extreme warrior society (though probably not entirely groundless) is still probably a bit of an overstatement. So for your question, an injured youth was probably medically treated about as well as in any other polis (which unfortunately, with their medical knowledge, wasn't too great. To paraphrase a saying of Oswyn Murray: if rationality is defined as coherence of ideas, Greek medicine is irrational whereas Greek religion is rational)

There was a really good article by Michael Flower on how the Spartan's themselves distorted their own past later on and may have made up many of these stories of an ultra militaristic society themselves. I'm sure it's in one of the many Hodkinson volumes though I can't remember which now.