Can a US Army Soldier wear the German Navy Schuzenschnur?

I have both the Army and Navy version and wear the Naval version on my uniform. The process was the same for the wear and approval for both (4187 up the chain to appropriate levels, translations, etc). I myself didn't know there was a difference until I saw a senior NCO wearing it in a shopette and inquired.
As of yet (having been awarded them 2015 and 2014 respectively), it has not been stated otherwise that it was not allowed, as it was approved and only one was used at a time. It has lead to confused looks and repetitive questions when appearing before promotion and quarter boards though, especially stateside.
Wings of victory and all that jazz.

Yes the only difference in the Army/ Airforce version and the Navy version is the color braid. the qualification standards are the same across all branches of the service with slight differences for Airborne and medical personnel based on the German version of the MTOE and weapons allocated to those MOS sets.
Depending on who sponsors the event Army, Airforce or Navy determines what color braid your badge will have at presentation as the qualifications and ranges are run by your partnership unit.

Why the question? He wasn't WEARING it, he had it on display. Just because a foreign country awards you something doesn't mean it authorized by DA or DOD for wear. No reg about showing it off!