Historians often dispute the claim but members of the Royal Navy itself generally believe it was originally created by King Alfred the Great in the 9th century...hence the 'royal'. (The actual phrase 'Royal Navy' did not appear anywhere in print before the 16th century, however.)
Of course, kings in history and throughout the world had armies long before they had navies, but these were generally recruited on an ad hoc basis in order to fight specific threats or campaigns. At the end of their necessity, they simply disbanded until next time. It is claimed, however, that there has been an 'established', permanent English - now British - navy since Alfred's day.
The RAF was formed at the end of World War I and given its 'royal' title at the time, having grown out of the Royal Flying Corps, when King George V was on the British throne.
Consequently, both the RN and RAF were specifically created in the name of a King and hence called �Royal'...the Army never was.