Assuming the statement in the original post is true and correct we can examine this by reference to the Lieber Code General Order 100. The Code was promulgated in April 1863 and this killing took place in December 1863. The Code specifies that:
Art. 56.
A prisoner of war is subject to no punishment for being a public enemy, nor is any revenge wreaked upon him by the intentional infliction of any suffering, or disgrace, by cruel imprisonment, want of food, by mutilation, death, or any other barbarity.
There are clauses in the code that allow for the execution of prisoners in several cases. These include executing prisoners found to have committed war crimes, execution as retaliation (presumably if the CSA executed Union prisoners), and where a soldier belonged to a unit which did not offer quarter. The OP is silent on these issues, but if the exceptions did not apply, then the execution was a war crime.
I do note that there is a limited, but vague, "military necessity" exception in the Lieber Code, as well as a lack of protection for guerrillas.