I've written on the history of the salute before which I'll repost here:

The Nazis did not invent it themselves. The 'Nazi Salute' was chosen because it was the 'Fascist Salute' already popularized by the Italian Fascist Party under Mussolini. Other Fascist aligned movements, such as the Spanish Falangists also adopted the gesture in similar imitation of the 'Ur-Fascist' group.

We'll return to the Nazis later, but now the first question to tackle is why did the Italians decide on the gesture? Mussolini was obsessed with creating a new Roman empire, and he adopted the trappings of 'Rome' as symbols for the party in furtherance of that obsession. The name itself, "Fascist", comes from the fasces which had once been a symbol of power and authority in Ancient Rome. The salute that they made use of, with the arm extended outwards, fingers together, palm down, was known as the "Roman Salute", so of course it was only appropriate that it would be the salute of the 'New Rome'. The association of the salute and a revitalized Roma-Italian Nationalist ideology predated Mussolini, who was likely influenced in picking it by the proto-Fascist thinker Gabriele D'Annunzio, who had implemented the salute during his short-lived control of the city of Fiume, and is the one who introduced it into the Italian nationalist lexicon.

But this still leaves us with another question to follow. Was the "Roman Salute" actually Roman? To which the answer is a fairly certain no! In no extant Roman sources or surviving Roman works of art is there representation of the salute that bears the name of Rome. We have evidence of salutes that involved raised hand but not in that manner - the closest examples, seen on Trajan's Column, have the fingers splayed out - and salutes not dissimilar to the modern military one as well. The famous statue Augustus of Prima Porta, although assumed by some to possibly be a Roman salute, almost certainly isn't. Aside from finger position, the simple fact is that the arm is a later restoration not original to the torso, and once upon a time the raised arm held a spear. Likewise, the Equestrian Statue of Marcus Aurelius on the Campidoglio can be erroneously identified as a Roman Salute, and was so even by the Fascists themselves, but it only works from very specific angles - close in, staring up - and is generally agreed by art historians to be a gesture of "benediction", one which is is much more easily identified as from many angles.

But if it isn't authentically Roman, what is it? In this regard, if there is any one, single culprit, it is certainly Oath of the Horatii, a late 18th-century neoclassical work showing a scene from Roman history of the Horatii, three brothers who triumphed in combat over the Curiatii, as they give their oath to their father prior to the combat. It is an evocative piece, considered a true masterpiece of the style, and greatly influential as a work of art:

The impact of the Oath was so revolutionary in the aesthetic realm that it radically altered the way artists made art and the way critics perceived it. [...] Modern historians have often referred to this as a pivotal work, one that signaled a decisive departure from the predominant, classicized Rococo style.

But its influence is beyond simple art, and for our purposes, it is the influence of the straight armed, fingers together, palm down salute that they render to their father accompanying the oath, although this of course ties into its aesthetics.

Unlike the actual Roman works where any similarities to the "Roman Salute" fall apart on examination, this one is unmistakable. It still isn't perfect, as they are at differing angles, and a mix of right and left arms, but the parallels, and the context, are unmistakable. Of course, the scene itself is a creation, a fanciful representation of filial and civic devotion that is not present in Roman accounts of the (mostly mythical) story, but that is quite secondary. Just as the scene is a creation, along with anachronistic clothing and weapons, the salute represented in the painting was chosen because of how it would impress the scene into the mind of the viewer, something which it is inarguably effective at. The work was possibly influenced by Roman images that showed Roman oath scenes where, weapon in hand, the soldiers pointed their swords downward, and as such the Roman oath motif generally was hardly alien to European art of the period, such as the Oath of Brutus by Gavin Hamilton that predated David's work by 20 years. But as Rosenblum put it, the style and gesture, including this new, specific presentation that was David's conception of the salute, left earlier works of "classical virtue [...] flaccid in both style and moral conviction". Although the caveat must of course be that this is art, so it is necessarily subjective, while Carrier certainly would disagree with such strong words, in his direct comparison of the two works noting the comparative "greatness" is not easy to answer, even is he agrees "the claim that Hamilton made a greater painting than David is unconvincing".

The specific degrees of 'greatest' though are not really our concern, insofar as they are secondary to the general tenor of critical acclaim, and especially how it relates to the use of gestures in the work. Much of the power of the painting is tied up the 'language of gesture' present in the work. It was, in fact, specifically the fact that David had chosen an essentially new gesture that helped get that across, since as Johnson notes of the debut:

The counterpoint of the right hand of the father is the group of swords clenched i n his left (an enormous feat of physical prowess), to which his sons swear allegiance with their pronated hands and by which two will perish . Most critics of the time were fascinated by the powerful impact of this pantomimic invention, which diverged so dramatically from well-known contemporary representations of antique, oath-taking scenes.

One such critic is quoted by her thus:

I will agree that it is a great conception and that it is executed as boldly as it is skillfully and I am as entranced as you are with the action of the Horatii, who embrace each other during their Oath, a sublime and symbolic expression of their union, of the sacred and courageous friendship that unites them, and of the common object that brings them closer and links them to one another until death, these three warrior brothers.

Later works would likely be influenced by David's powerful use of body language and gestures, such as Jean-Léon Gérôme's The Death of Caesar, where the arms of the tyrannicides, thrown upwards weapons clasped, possibly draws from The Oath, "express[ing] a kind of reaffirmation of their allegiance, a renewal, as it were, of their sworn brotherhood." Certainly David himself knew that he had struck something good, as the same gesture shows up in his later The Tennis Court Oath, a work depicting the titular event in the French Revolution, and almost certainly did so not only because of the power of the gesture itself, but to consciously connect his two works, and the connection between both the ancient and modern oath. By 1810, and his work The Distribution of the Eagle Standards, hopefully you're noticing the theme here, namely that he knew to milk this thing for all its worth, but more importantly it should be noted that the meaning is expanding, and the oath is more imperial, Napoleon's military commanders showing him their loyalty. And although obviously a French scene, the gesture, through its genesis, was now a Roman one - Napoleon too going back to that old Empire for symbolism - and it is impossible to not envision the composition of the work unchanged, and only the figures transposed, with Italian Black Shirts, or Nazi SS men taking the place of the French soldiery. For this scene, which shows an event three days after the Emperor's coronation when the regimental commanders came to swear their oath of loyalty to him, as Boime poignantly notes the shift:

The series of oath pictures may be seen as the coding of key developments in the history of the Revolution and its culmination in Napoleonic authoritarianism. [...] The civil pride of French nationalism won during the Revolution had been displaced onto pride in battlefield glory, and the welfare of the French citizenry taken as a whole became subordinated to the prestige of the troops. Symbolically this was further represented by shifting the ancient paradigm from the republic to the empire.

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