The former England captain started his career at Old Trafford and won the FA Youth Cup before being integrated into the first-team setup by the Scot, who will step down at the end of the season, with David Moyes set to replace him.
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"I've just seen the news on Sir Alex," Beckham wrote on his Facebook page. "As I have said many times before, the boss wasn't just the greatest and best manager I ever played under, he was also a father figure to me from the moment I arrived at the club at the age of 11 until the day I left.
"Without him I would never have achieved what I have done in my career. He understood how important it was to play for your country and he knew how much it meant to me.
"After [World Cup] 1998, without the manager I would have found it virtually impossible to cope with the attention I was getting on and off the field and for this I will always be grateful to him for his support and protection.
"I am truly honoured to have been guided by the greatest manager in football and to have had the career that I had under him.
"Thank you boss and enjoy the rest!"