(For an answer to the actual question (i.e. why do we call him that), see below.)
I spot two possible factors for why the “R. R.” version of his name is a better choice:
1. It sounds better
Martin himself noted in an interview that as an author, your name is your brand, and therefore having a memorable and unique one like “George R. R. Martin” is advantageous.
He states that his second middle initial comes from his confirmation name “Richard”, which he chose for himself at age 13 (in 1961/2) because his cousin was called Richard.
However — and this is pure speculation — Wikipedia notes that Martin was already writing (and selling!) somewhat mythical stories around this age, and was a voracious reader. Given that Lord of the Rings was published in 1954, it’s possible that J. R. R. Tolkien’s middle initials had some influence over this choice (though I haven’t found any actual evidence to support this).
At the very least, when aiming to successfully sell a complex series of fantasy novels involving several interwoven mythical storylines about kings, battles, monsters and destiny, it’s difficult to believe that no-one at Martin’s publishing company ever raised the idea that having your middle initials be “R. R.” could lead to some positive brand associations. (Although Martin was already publishing under the name “George R. R. Martin” long before the first novel in the Song of Ice and Fire series was published.)
2. It avoids disappointing Beatles fans
As Mr Lister noted, George Martin is the name of a pretty famous record producer who signed The Beatles and was closely involved in making their records.
Although it’s not really in the same field, or indeed the same era of popular culture, in the age of Google the top result is everything, and second place is nowhere .