Like us, the Nazis took this salute from the ancient ROMANS.
American school kids were still using this method to salute the American flag in the late 1930s. When it became known that the Nazis were also using it, Americans changed and began placing their right hand over their heart for the pledge of allegiance.
While the Nazis have forever tarnished that salute (much like how the Confederate flag is suddenly seen today as a symbol of hatred & racism), they certainly did not steal it from us.