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.

Simmons, 52, was suspended in September for questioning the continued omissions of Dwayne Bravo and Kieron Pollard.
He issued a public apology to the West Indies Cricket Board (WICB).
"I regret my public outburst during a press conference and the revelation of the vote that took place at the selection meeting," said Simmons.
"I extend a public apology to all whom I may have offended. I have personally apologised to my fellow selectors and the WICB and now do so again publicly.
"I am looking forward to putting this behind me and doing the job I enjoy with the West Indies' senior men's teams."
Simmons played 26 Tests as an all-rounder for the West Indies between 1988 and 1997, and became head coach of their senior squads in March.
Eldine Baptiste took over from Simmons for the current tour of Sri Lanka, with the West Indies losing both Tests and all three one-day internationals before Wednesday's concluding Twenty20 match.

Ms Tiffney, 43, was last seen leaving her home in Edinburgh's Dean Village in May 2002.
The remains were found by a cyclist in an area off the A198, near the entrance to Gosford House in Longniddry on Sunday evening.
A murder charge against Ms Tiffney's son Sean Flynn was found not proven at the High Court in Perth in 2005.
Police said her death remained a murder investigation and that detectives would continue to liaise with the Crown Office and procurator fiscal service.
Police Scotland said there was no reason to speak to Mr Flynn at this stage.
They said the area, about eight metres from the roadside, where Ms Tiffney was found had not been searched before.
Det Ch Insp Keith Hardie, of Police Scotland, said: "Now that we have identified these remains, our thoughts are once again with Louise's family, who have had to wait 15 years for this news.
"We will continue to provide them with all the necessary support and assistance they may require at this very difficult time.
"It is not as bad news for the family as it would have been at the time.
"I think they had given up any hope of finding Louise alive so I suppose, to a certain extent, it's good news for them because there is the possibility of bringing this to some form of conclusion."
"Our investigation into Louise's death and the discovery of her remains is continuing and we will keep the family and the Crown Office fully updated with the progress of these inquiries."
The identity of the remains was confirmed using dental analysis.
However, police said they would also seek corroboration through DNA.
The force added that Ms Tiffney's remains had now been fully recovered from the site.
The A198, which had been closed since the discovery, was reopened on Friday.
In June 2002, Mr Flynn was given three years and nine months in a young offenders' institution after admitting causing the deaths of two friends in a car crash.
Three years later, he stood trial accused of Ms Tiffney's murder.
His trial heard that while waiting to be sentenced over the fatal crash, he and his mother quarrelled at their home.
The Crown alleged that he killed her and hid her body.
However, a jury at the High Court in Perth returned a verdict of not proven.
Mr Flynn expressed his relief, but relatives of Ms Tiffney reacted with fury to the verdict.
The Duke of Kent has left hospital following successful treatment for a dislocated hip.
[NEXT_CONCEPT]
West Indies have reinstated Phil Simmons as head coach of their senior squads after he apologised for criticising team selection.
[NEXT_CONCEPT]
Human remains found in East Lothian are those of missing woman Louise Tiffney, police have confirmed.