If a fellow soldier is injured by the reckless driving of another soldier while on duty, can the court martial award monetary compensation to the injured soldier from the defendant?

I don't know the regulation but I hope not, it's a criminal matter, not a civil suit. If you want money to repair your vehicle or something then why don't you just take him to court yourself? If you have a liable reason for requesting the money and he was truly responsible for the accident then you can also request to have him pay your attorney fees and court fees.

Doesnt the other Soldier have TRICARE to cover the costs of any medical expenses?

Tricare will go after the party at fault whenever a soldier goes to the emergency room for a vehicle accident. Tricare is free to the military member but still costs the taxpayers money. I have seen them go after insurance companies for payment whenever a soldier or family member is in a not-at-fault accident. But, to answer the first question, NO, UCMJ will not pay the soldier for any sort of compensation. Like another member posted, a civil suit would have to be filed in order to receive compensation or if both drivers were insured, then the insurance companies would deal with the compensation. Make sure that the soldier who was not at fault gets a good injury lawyer, this can help tremendously when it comes to how much the insurance company will pay.