The 38-year-old, who departed United in 2003, made just short of 400 appearances under Ferguson, scoring 85 goals in the process.
"I've just seen the news on Sir Alex. 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," Beckham said on Facebook.
"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 '98 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!"
Beckham's current deal with Paris Saint-Germain will expire at the end of the season.