Mr Starr posted a black and white picture of the band with Mr Martin and wrote:
Thank you for all your love and kindness George peace and love xx pic.twitter.com/um2hRFB7qF— #RingoStarr (@ringostarrmusic) March 9, 2016
British Prime Minister David Cameron led tributes to the producer, who signed up The Beatles when he was the head of the Parlophone label after hearing their demo record in 1962. (Also Read: George Martin Dies at 90. Twitter Bids Farewell to 'Fifth Beatle')
Sir George Martin was a giant of music - working with the Fab Four to create the world's most enduring pop music.— David Cameron (@David_Cameron) March 9, 2016
The producer, who signed up the Beatles after hearing their demo record in 1962, was so essential to the band's success that he was nicknamed the "Fifth Beatle".
Late band member John Lennon's son Sean wrote on Instagram:
Mr Martin was born in January 1926, a carpenter's son from north London.
After serving in the Second World War, he studied at the prestigious Guildhall School of Music and began playing the oboe in bars and clubs around London.
His first job was in the BBC's music library.
He then joined the record label Parlophone, a division of EMI, and rose to become its head by 1955 at the age of just 29.
First single 'pretty poor'
Mr Martin quickly realised The Beatles' potential after first hearing them in 1962.
"I liked them as people apart from anything else and I was convinced that we had the makings of a hit group," he told Melody Maker magazine in an interview. But he added that: "As composers, they didn't rate. They hadn't shown me that they could write anything at all."
The band's first single Love Me Do, which reached number four in the charts in October 1962, "was pretty poor but it was the best we could do," he said.
That song was followed by Please Please Me, which made number two, and From Me To You in April 1963 - their first chart-topper.
Mr Martin also composed scores for The Beatles films A Hard Day's Night and Yellow Submarine.
After the band split, he built the Air Studios on the Caribbean island of Montserrat, which was forced to close in 1989 due to damage sustained by Hurricane Hugo. He went on to work with other artists including Bob Dylan, Sting and Elton John.
He also recorded two of former Beatle Paul McCartney's solo albums, Tug of War and Pipes of Peace.
He continued to write music and work with music charities until his death.
Music stars reacted quickly to the news, with singer Lenny Kravitz paying tribute to Mr Martin as a "visionary producer" and tweeting:
Oasis frontman Liam Gallagher tweeted:
Sir George Martin RIP LG x— Liam Gallagher (@liamgallagher) March 9, 2016