"He was the kindest, coolest, most patient and loving father. He always put family before self," Peter wrote in a statement on behalf of his family.
Peter added: "He looked forward to any chance to connect with his grandchildren, and had a unique bond with each of them. He and his youngest grandson, Whit, nine, could often be found in the corner of a room at family parties having deep philosophical conversations."
McCallum was known as "a true renaissance man… fascinated by science and culture, and would turn those passions into knowledge", the statement added.
Peter added that his father was "capable of conducting a symphony orchestra and (if needed) could actually perform an autopsy, based on his decades-long studies for his role on NCIS".
"After returning from the hospital to their apartment, I asked my mother if she was OK before she went to sleep. Her answer was simply, 'Yes. But I do wish we had had a chance to grow old together.' She is 79, and dad just turned 90."
Peter continued: "The honesty in that emotion shows how vibrant their beautiful relationship and daily lives were, and that somehow, even at 90, Daddy never grew old."
McCallum made his screen debut in 1953 in the BBC fantasy miniseries The Rose and the Ring.
He went on to appear in other TV shows including a 1959 BBC adaptation of Jean Anouilh’s Antigone, as well as BBC adaptations of Jane Austen’s Emma in 1960 and Wuthering Heights in 1962.
Over the course of his acting career, McCallum also appeared in a number of films including A Night to Remember, The Great Escape and The Greatest Story Ever Told.
But it was his role as Illya Kuryakin in the hit 1960s spy drama The Man From UNCLE that catapulted McCallum to fame, as well as for his part as pathologist Dr Donald 'Ducky' Mallard in NCIS.
Alongside his acting career, McCallum recorded four albums for Capitol records in the 1960s, and also released his own crime novel in 2016, titled Once a Crooked Man.
McCallum is survived by his wife of 56 years, Katherine McCallum, as well as his sons Paul, Valentine and Peter, his daughter Sophie and his eight grandchildren.