The Queen's cousin, 79, was taken to Aberdeen Royal Infirmary on Monday after he suffered the injury while staying at Balmoral.
He was seen walking to a waiting car with the help of a walking stick.
The duke, who undertakes a large number of official royal engagements and has close associations with many charities, suffered a "mild stroke" in 2013.
He is perhaps best known as president of the All England Lawn Tennis Club, in which capacity he has presented trophies to winners at Wimbledon.
He is also grand master of the main organisation of Freemasons in England, the United Grand Lodge.
He became Duke of Kent in 1942 at the age of six after his father, brother of King George VI, died in an RAF plane crash.
Balmoral is the Scottish home of the Royal Family.
The Duke of Kent has left hospital following successful treatment for a dislocated hip.