Magdy Abdel Ghaffar also said that authorities were exerting "huge efforts" to investigate last month's torture and murder of Italian student Giulio Regeni, which media in Italy suspect to have been the work of Egyptian security services.
"We apologise to every citizen who has been insulted," the minister said at a joint press conference with Prime Minister Sharif Ismail broadcast live on state television.
"We apologise for the acts of some policemen, we kiss the head of every citizen subjected to abuse or insult or any unkind act by policemen," he said, in a rare apology by Egyptian authorities.
Policeman Mustafa Mahmud is alleged to have shot dead driver Mohamed Ismail with his police-issued firearm late Thursday in central Cairo after a row over the fare for a delivery.
The killing triggered outrage on social media and dozens of people staged a protest outside Cairo police headquarters.
A day after the shooting, President Abdel Fattah al-Sisi said he would ask parliament to amend the law to toughen sanctions for police abuses.
Police abuse was a key factor that sparked the 18-day uprising that toppled longtime president Hosni Mubarak in 2011.
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Five years on, human rights groups are again denouncing the police force after a string of torture and deaths in police stations, arbitrary arrests and the disappearances of Sisi regime opponents.
Abdel Ghaffar said the authorities were exerting "huge efforts to reveal (circumstances of) the mysterious" death of Regeni, a Cambridge University PhD student from Italy who disappeared on January 25 in Cairo.
His body was found with signs of torture on February 3 in a ditch in a suburb of the Egyptian capital.
Italian media has charged that Egyptian security services were involved in his death, a charge rejected by Cairo.
Prime Minister Ismail said the image of the police needed to be "transformed" in the eyes of Egyptians.
"There will be no laxity and we will not cover up any (violation)" by the police, he said, standing alongside Abdel Ghaffar.
"This is a commitment from the police, a firm commitment. If there is any violation from some individuals, they will be held accountable."
Over the past year, several policeman have been detained for violence against prisoners and some have been sentenced to jail terms.