Our armed forces practice of rendering a hand salute by raising an open hand to the brow can be traced back through Great Britain to medieval knights. Although, an ancient tradition, this salute has evolved and recent nuances have adapted it to modern time.
Prior to jousting in tournaments, knights would raise their visors to prove their identities and show respect. This evolved in the British army to doffing one's hat bat, which became impractical to touching the cap or brow. Currently, the Royal Air Force and Royal Army render the salute with the palms famously forward conversely, Royal sailors salute with their palms down so as to not offend officers with their dirty blistered hands. Our saluting practice is based on that of the Royal Navy - palms down.
My viewpoint on this subject is tainted since I am a retired USMCR Captain. Starting with officer candidate school, I was drilled in the importance of delivering a proper hand salute. Which is when you touched our brow or hat brim with your fingertips with your upper arm parallel to the deck with your forearm and hand forming a straight line then returning your hand to your side. This is all done in a 'snappy manner' as in snap and pop. While the Air Force and Navy have slight variations, the U.S. Army salute appears to be the same as that of the USMC.
In a few days we will have a new President. American Presidents have always been saluted, but it is only recently that Presidents have rendered the military style salute. The following U.S. Army Generals did not continue to salute in a military fashion as President; George Washington, Andrew Jackson, William Harrison, Zachary Taylor, Franklin Pierce, Ulysses Grant, Rutherford Hayes, James Garfield, Chester Arther, and Dwight Eisenhower. These Presidents were comfortable with their military bona fides and did not continue to slate like a soldier upon ascending a civilian office. I have no doubt that President Trump will salute in a military style.
Contemporary historians credit Ronald Reagan with starting the current trend of civilian Presidents saluting like soldiers. President Reagan consulted with the foremost on saluting, the commandant of the Marine Corps, General Robert Barrow and was also advised, "Your're the #@$%ing commander-in-chief you can #@$%ing do whatever the #@$% you want." General Barrow was also known for his fluency in Navy lingo.
When I see President Obama salute, I am impressed - he salutes like a Marine. Other Presidents have used different styles reflecting their military service. Recent Presidents showed some, swat the fly (Navy) or, thumb your nose (Air Force influence, without prior military experience I expect President Trump to salute like Obama who also has no military background.
Regardless of the President's salute he should be saluted by virtue of the office.