Probably not. This site gives a decent explanation in brief, but the short version is that the US military customs and courtesies generally evolved out of those of their British forebears. Removal of the hat was traditional, yet cumbersome, so the Brits switched to the backhand salute we still see. Not sure when the US salute diverged, but that's its origin, unless someone more well-informed comes along. This page has a WWI picture of American soldiers saluting, which more directly answers your question.

Is it true that the U.S.' original military salute, predating World War II, was the same as the Nazi's military salute?

I think you may be confusing that with the fact the the Cub Scout's salute was changed because it looked somewhat like the Nazi salute.

From the Boy Scouts of America:

The Cub Scout sign (the index and middle fingers extended in a V shape) symbolizes the ears of an alert wolf. It replaced the Indian “how” sign, which looked too much like the Nazi salute.

The US Military salute has remained largely unchanged since the 1820s. OP may be thinking of the Bellamy Salute, associated with the Pledge of Allegiance.