The Spartacus star passed away on 5 February 2020 aged 103 as one of the last surviving stars of cinema's Golden Age.
"Can’t believe it’s been a year since you left us," Michael wrote alongside an Instagram photo of himself with his late father. "At 103, you picked a good time to check out. I love you with all my heart! #KirkDouglas."
A tribute was also paid by his daughter-in-law Catherine Zeta-Jones who wrote in the comments of husband Michael's post: "Love you Pappy🙏🏻😘."
On his father's passing last year, Michael said in a statement: "To the world he was a legend... but to me and my brothers ... he was simply Dad.
"Kirk's life was well lived, and he leaves a legacy in film that will endure for generations to come, and a history as a renowned philanthropist who worked to aid the public and bring peace to the planet."
In addition to Michael, he was survived by his younger son from his first marriage to Diana Dill, 74-year-old Joel as well as his second wife Anne Buydens, 101, and their son Peter, 65. Their other son, Eric, died in 2004 at the age of 46 following an accidental overdose of drugs and alcohol.
The Douglas family recently added a new member as Michael's eldest son Cameron, whose mother is his first wife Diandra Luker, welcomed a son.
The 42-year-old's partner Viviane Thibes gave birth to a baby boy named Ryder toward the end of 2020. The couple are already parents to three-year-old Lua.
Watch: Michael Douglas meets his first grandson