The Germans were violating international law. The crime the Americans committed was not because of execution per se, but because of summary execution.

The law at the time was the Laws and Customs of War on Land (Hague II); July 29, 1899. Article 1 of the main body requires "The laws, rights, and duties of war apply not only to armies, but also to militia and volunteer corps, fulfilling the following conditions: ... To have a fixed distinctive emblem recognizable at a distance ..."

Article 23 is most applicable: "Besides the prohibitions provided by special Conventions, it is especially prohibited:-- ... To make improper use of a flag of truce, the national flag, or military ensigns and the enemy's uniform ..."

Chapter II, being articles articles 29-31, is "On Spies". "An individual can only be considered a spy if, acting clandestinely, or on false pretences, he obtains, or seeks to obtain information in the zone of operations of a belligerent, with the intention of communicating it to the hostile party. ... A spy taken in the act cannot be punished without previous trial."

So technically you'd have to prove that the Germans were there to gather intelligence; just wearing the uniform is just "especially prohibited". And execution without trial is prohibited. But the Germans were in the wrong in the first place, and given the odds of a prisoner being illegally killed on the spot even in normal circumstances (I think 50%ish), it's unsurprising.

Summary execution of a spy is a crime under international law. Spies may be executed after trial under the laws of the capturing force if those laws provide for execution and if the offense is proved. The history behind spies being treated differently than uniformed soldiers is provided in answer to an earlier, similar question.

https://www.reddit.com/r/AskHistorians/comments/2k6z06/how_did_the_different_treatment_of_spies/