Gregory died at hospital in Washington DC after being admitted with an undisclosed illness.
The family shared the news of their "enormous sadness" in a post on social media on Saturday.
From the 1960s Gregory mocked racism in his stand-up routines. He was among the first US black comedian to perform regularly in front of white audiences.
"It is with enormous sadness that the Gregory family confirms that their father, comedic legend and civil rights activist Mr Dick Gregory departed this earth tonight in Washington, DC," his son, Christian, wrote on Instagram.
He added that the family "appreciates the outpouring of support and love" but requested that their privacy be respected "during this very difficult time".
US civil rights activist Jesse Jackson paid tribute to Gregory on Twitter:
End of Twitter post  by @RevJJackson
Gregory was a no-holds-barred comedian, delivering lines in his performances that were likely to raise eyebrows.
"A Southern liberal?" he once said, "that's a guy that'll lynch you from a low tree".
"Segregation is not all bad," Gregory said on another occasion. "Have you ever heard of a collision where the people in the back of the bus got hurt?"
Gregory was also involved in a number of civil rights activities in the 1960s. He campaigned against the Vietnam War and his protests often involved hunger strikes.

The painting was revealed by the veteran broadcaster and naturalist at New Walk Museum, in Leicester.
He grew up in the city and volunteered at the museum as a teenager.
The artist - Bryan Organ - has also painted Sir David's brother Lord Attenborough, Prince Charles and former prime minister Harold Macmillan.
The portrait was commissioned by the City of Leicester Museums Trust and will complement the portrait of Lord Attenborough, the acclaimed actor and film director, which also hangs in New Walk Museum.
More on this and other Leicestershire stories
Sir David said it was "an honour" to sit for Mr Organ.
When asked what he made of the portrait, he said: "I haven't had the chance to have a look at it - I mean, I've only seen it obliquely. As far as I can see, it's very nice.
"It was great fun sitting with Bryan, the only problem was the painting got in the way."
The Attenborough brothers grew up in the city where their father was principal of the then University College Leicester.
Tributes have been paid to Sir David throughout the year since he turned 90 in May.

Mr Jammeh has denied similar allegations in the past.
He will be seeking re-election for a fifth term and for the first time he will be facing a single major opposition candidate.
HRW urges external pressure on The Gambia to prevent further violence.
In a report published on Wednesday, the group says two Gambian activists have died in custody, with dozens of critics of President Jammeh in jail.
They are being denied medical or legal help, the report says.
The Gambia, a popular tourist destination because of its beaches, has never experienced a democratic transition of power since independence.
Mr Jammeh took over in a 1994 coup. His critics accuse him of running a repressive state, an allegation he denies.
Most of the major Gambian opposition parties last week picked a single candidate - businessman Adama Barrow of the United Democratic Party (UDP).
Mr Barrow's nomination is the result of the biggest opposition alliance in Gambia since independence, the AFP news agency reported.
HRW says the prospect of political change in December has prompted Mr Jammeh's government to step up oppression against critics.
It claims a torture squad, linked to the presidential guard, is terrorising opponents ahead of the election.
The group wants freedom benchmarks to be imposed now - by the United States, the European Union and The Gambia's West African neighbours - to prevent more bloodshed.
Last week, The Gambia said it would pull of the International Criminal Court (ICC) after accusing the tribunal of persecuting and humiliating Africans.
The small West African nation joined South Africa and Burundi in saying it will withdraw from the court.
US comedian and civil rights activist Dick Gregory has died at the age of 84, his family has announced.
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A portrait of Sir David Attenborough has been unveiled to commemorate his 90th birthday - in a museum where he once volunteered.
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President Yahya Jammeh of The Gambia is using violence to silence critics in the lead-up to presidential election in December and pressure is needed to stop it, Human Rights Watch (HRW) says.