"To the world, he was a legend, an actor from the golden age of movies... but to me and my brothers Joel and Peter he was simply Dad," it read in part
"Let me end with the words I told him on his last birthday and which will always remain true. Dad - I love you so much and I am so proud to be your son."
Michael's wife and Kirk's daughter-in-law, Welsh actress Catherine Zeta Jones, posted a photo of the two together, writing: "I shall love you for the rest of my life. I miss you already."
An icon who defied convention
Analysis by Peter Bowes, BBC News, Los Angeles
When I met Kirk Douglas in 2008 he was a sprightly 91. He talked about his advancing years and the impact a stroke, in 1996, had on his skills as an actor.
"I couldn't talk at all," he told me. "So what does an actor do who can't talk? He waits for silent pictures to come back! That's a corny joke," he chuckled.
Douglas was particularly proud of his role in ending the Hollywood blacklist, when he defied the ban on working with filmmakers with alleged communist ties or sympathies.
He said he drew on "the impulsive qualities of younger Kirk" in making his decision to give the blacklisted writer Dalton Trumbo a screen credit under his own name for his work on Spartacus.
We discussed his passion for working with young people. He had started writing a blog to encourage young Americans to vote in that year's presidential election.
Douglas and his wife donated millions of dollars to charitable causes and helped build hundreds of school playgrounds. He said their philosophy was: "Before you die, try to do something for other people."
Douglas was prolific as a film actor, with more than 90 credits to his name - ranging from the 1940s to the 2000s.
He is perhaps best-known for Spartacus, a Stanley Kubrick film which won four Oscars and was so popular that its iconic "I am Spartacus" scene entered the pop cultural lexicon.
Douglas was himself nominated for an Oscar three times - for Champion (1949), The Bad and the Beautiful (1952), and Lust for Life (1956). He eventually won the honorary award in 1996 in recognition of his 50 years in the industry.
He faced difficulties in his personal life. He narrowly survived a helicopter crash in 1991 that left two people dead. Five years later, he suffered a major stroke that affected his speech.
And in 2004, his son Eric died at the age of 46 of an accidental drug overdose.
In his later years, he turned his attention to charity. He donated millions of dollars to charitable causes and helped fund an Alzheimer's unit at a retirement home in Los Angeles.