It appears that the chant comes from the Howard Beach racial incidents of 1986-87 and was first popularized by Sonny Carson (1936-2002), not Sharpton.
"No justice, no peace, no racist police!" has been cited in print since at least 1995.
Wikipedia: Al Sharpton
Alfred Charles "Al" Sharpton, Jr. (born October 3, 1954) is an American Baptist minister, civil rights activist, and television/radio talk show host
On December 20, 1986 three African-American men were assaulted in the Howard Beach neighborhood of Queens by a mob of white men. The three men were chased by their attackers onto the Belt Parkway, where one of them, Michael Griffith, was struck and killed by a passing motorist.
A week later, on December 27, Sharpton led 1,200 demonstrators on a march through the streets of Howard Beach. Residents of the neighborhood, who were overwhelmingly white, screamed racial epithets at the protesters, who were largely black. A special prosecutor was appointed by New York Governor Mario Cuomo after the two surviving victims refused to co-operate with the Queens district attorney. Sharpton's role in the case helped propel him to national prominence.
22 January 1987. St. Petersburg (FL) Times, "4,500 march to protest racial attack in N.Y." by Dan Jacobsen, United Press International, pg. 11A:
NEW YORK - More than 4,500 black protesters chanting slogans to the beat of pounding drums marched in Manhattan Wednesday during a day-long demonstration of "outrage" against the Howard Beach racial attack.
With fists clenched in anger, they filled Broadway at 32nd Street, site of a city welfare hotel, then led a deafening demonstration down Fifth Avenue to Mayor Edward Koch's Greenwich Village home.
Chants of "No justice, no peace" and "Mayor Koch step aside, there ain't gonna be no genocide" echoed among the office buildings as police scrambled to line the route.
22 January 1987, Newsday (Long Island, NY), "4,000 March Against Racism But Impact Of Boycott Less Clear," pg. 3:
To the deafening beat of chants, a predominately black army of nearly 4,000 protesters marched down Fifth Avenue yesterday in a declared effort to defeat the notion that blacks will tolerate racial injustice.
The march highlighted "The Day of Mourning and Outrage" for Michael Griffith, who was killed by a car Dec. 20 while fleeing a gang of white teenagers in Howard Beach, Queens.
Chanting "No justice, no peace," the five-block-long stream of marchers, some shaking clenched fists, set off from the Martinique Hotel, a welfare hotel on West 32nd Street. They marched to Mayor Edward I. Koch's Greenwich Village home, a 30-block walk that took an hour.
12 February 1987. Newsday (Long Island, NY), "Black Leaders Say Charges Just a Start" by Patricia Hurtado, pg. 26:
Yesterday, nearly two dozen speakers took the podium. One speaker who received of the most rousing response from the audience was black activist Sonny Carson, who promised a rebirth of black militancy.
"No justice! No peace!" Carson shouted. "No peace for all of you who dare kill our children if they come into your neighborhood . . . We are going to make one long, hot summer out here . . . get ready for a new black in this city!
6 July 1987, New York (NY) Times, pg. 35:
"No justice, no peace," said Mr. Carson repeatedly in what he said he hopes will emerge as the rallying cry for his cause.
New York City • Government/Law/Politics/Military • Thursday, May 26, 2005 • Permalink