Edwards, 25, had drawn his gun and was chasing a man who had broken into his car. Dunton apparently mistook the off-duty officer for an armed suspect and shot him.
Since then, some civil rights advocates have charged that race was a factor in the shooting.
The Rev. Al Sharpton said Thursday that the decision was expected, as grand juries rarely indict officers in friendly fire shootings.
"We will continue to call on the governor to authorize a special prosecutor in this case," Sharpton said.
Edwards' mother, Natalia Harding, said in a television interview that she believed Dunton shot her son because he was black.
"I would like to see him go to jail," Harding said.
Dunton, 30, has been on administrative duty since the shooting. Police officials said the grand jury decision clears the way for an administrative review of the shooting to see if it fell within department guidelines for use of deadly force.
A spokesman for the police union declined comment Thursday.
The district attorney said Edwards was in street clothes and was walking to his car in East Harlem when he noticed the driver's-side window was broken and a man was rifling items on the front seat.
Edwards drew his gun and grabbed the man's shoulder with his left hand, Morgenthau said. A struggle ensued, and the man ran. Edwards chased him, gun in hand.
Dunton and two other plainclothes officers, in an unmarked car, spotted the two men. They saw that the pursuer had a gun in his hand and pulled over.
According to the district attorney, Dunton got out of the car and yelled, "Police, don't move. Drop the gun. Drop the gun."
Edwards slowed but did not stop, turned and pointed his gun at Dunton, Morgenthau said.
Dunton fired six shots, mortally wounding Edwards. The man Edwards was chasing was arrested and charged in the case, police said.
Emergency service officers arrived and tried to revive Edwards. It was only when they cut his sweatshirt open and saw his Police Academy T-shirt that they realized he was an officer.
In a statement Thursday, the New York Police Department outlined several precautions ordered by Commissioner Raymond Kelly that seek to minimize the risk of friendly fire.
They include hiring a psychology professor to study shooters' decision-making and surveying undercover officers for advice.
In addition, the NYPD is studying new technologies that could avert friendly fire shootings. One early warning system would equip officers with devices that, once their guns are drawn, emit a signal to other officers in the area.
Mayor Michael Bloomberg told mourners at Edwards' funeral: "I promise you, we will do everything possible to learn from this tragedy."
Associated Press writer Tom Hays contributed to this report.