The king who is known to have had a speech impediment was King George VI, father to the present Queen, who reigned from 1936 to 1952. The matter of his speech impediment was dramatised in the film The King's Speech (2010) written by David Seidler, in which Colin Firth plays the part of the King. This clearly has nothing to do with the formation of the Received Pronunciation which was already long established before George VI was born.
I have heard it said, but have never read anything about the matter, that one of the earlier Georges, possibly George II (reigned 1727 - 1760), who had difficulty with English (he had been born a German, and German was his first language) had rather quaint ways of saying things. Some of his more idiosyncratic expressions were affected by courtiers and other sycophants - which may explain a few surviving odd-sounding British expressions such as What ho and Hey what.
But it would be ridiculous to suppose that the entire system of Received Pronunciation could have become established in that way.
Note on rhotacism
Rhotacism is a speech impediment involving difficulty in pronouncing the letter r. It probably has nothing whatever to do with the rhotic r sound.
Most regional dialects in Britain do not sound the rhotic r. It is however a feature of the various West Country accents, from Cornwall in the south-west to Hampshire and Berkshire in the east. One theory holds that it is the origin of the North American rhotic r.