Some historical background: the so-called "Roman salute" does not appear to have ever been used in Ancient Rome. No clear example of it is found in existing artifacts from either Republican Rome or the Roman Empire.
Rather, the gesture now called the "Roman salute" appears to have been invented by Jaques-Louis David in his 1784 painting entitled
The Oath of the Horaitii. Though apparently historically inaccurate, it was afterwards used in many neoclassical works of art dealing with Ancient Rome. The gesture wasn't explicitly associated with European Fascism until Gabriel D'Annunzio made the association of that gesture with Italian Fascism in the early 1920s. Prior to then, it was simply an invented gesture inaccurately associated with Ancient Roman culture.
In fact, outside artistic circles the so-called Roman salute appears to have been largely unknown in the US by that name until accounts/photos/films of its use by European Fascists began to appear in news media in the 1920s.
However, a similar gesture - the
Bellamy salute - was in fact apparently independently invented in the 1890s and thereafter used in the US during recital of the Pledge of Allegiance. Due to the similarity of that gesture (the Bellamy salute) to the Fascist salutes used prior to and during World War II, the US Flag Code was changed in 1942 to specify the "hand over heart while reciting the pledge" (that many Americans alive today remember) instead of the Bellamy salute.