This was actually due to a directive handed down by joint American/British SHAEF command (Supreme Headquarters Allied Expeditionary Force) in, Handbook Governing Policy and Procedure for the Military Occupation, for the Western Allied Military Occupation of Germany.
The decision was made by planning staff prior to the occupation that in order to help dismantle the military caste in Germany, saluting and military courtesies would be heavily curtailed. Especially when taking place between Allied and German soldiers during the occupation.
The directive originally called for no salutes to be exchanged between Allied and German soldiers, but was then revised so as to allow Allied officers (and enlisted if the case warranted it) only to return salutes initiated by German officers/enlisted when they were acknowledging orders given to them by Allied officers. Otherwise, German soldiers were to only come to attention in the presence of Allied officers and personal (unless they were in movement outdoors). German soldiers could however, salute each other (though civilians were barred from saluting them) during the occupation, so long as it wasn't with the Nazi salute.