He would probably go on living the life of a leisured gentleman, just like all other Spartan citizens.

His status as a Spartiate did not depend on his ability to fight. He was a Spartiate because he was born to Spartiate parents, and because he owned enough property to be able to pay his contributions to the common mess of which he was a member. As a citizen, he had to serve in the militia from age 20-60, but men over 60 did not lose any rights just because they were too old to fight.

The high property requirement for Spartan citizenship meant that all Spartiates were, by definition, members of the leisure class. Like all wealthy Greeks, they enjoyed spending their spare time hunting, raising and riding horses, dancing, and engaging in common meals and drinking parties. While our one-handed Spartan might have a bit of trouble with all these things, he'd have a lot more time to spend now that he was excused from exercises with his messmates, and he would presumably pick up a less taxing hobby of some sort.

There were only two ways a Spartan could lose some or all of his rights as a citizen. One, famously, was if he was considered a coward (a tresantes, "trembler") for running away from battle. These men were supposed to be publically shunned; they had to dress in rags and shave half of their beard so that everyone could see their disgrace, and - Xenophon poignantly adds - no one would play ball games with them. If our Spartan lost his hand in a battle the Spartans were losing, he might face the accusation of being a coward. However, our sources give us only one actual example of someone being subjected to this treatment. There are several examples of this law being deliberately ignored. Even if he ran away after losing his hand, our Spartan may not have lost any of his citizen rights.

The other way, which very certainly was ruthlessly enforced, was if he fell below the property level required to be a Spartiate. If he wasn't rich enough, he was out. However, there's little reason for our one-handed Spartan to fear this, unless he had a daughter's dowry to pay. Since Spartiates weren't allowed to have any profession and ideally weren't supposed to do any work, the loss of a hand shouldn't influence his income too much. It was never the product of his own hands to begin with. His wealth was based on his ownership of land worked by state serfs called helots.

In short, while the loss of a hand would no doubt have caused our Spartan some shame and agony, in the end it might make his life a lot easier. He would no longer be subject to the rigorous exercise programme that all Spartiates were meant to take part in, and he would no longer be liable for military service when the militia was called up. Meanwhile he would still be able to sit in the Assembly, participate in the dinners of his messmates, and generally live the good life of a leisured Greek gentleman. A million-dollar wound!