Published Thursday, 27 February 2014
Mr Kelly says he wanted to speak to the driver. (© Sinn Féin)
It happened during a disturbance in the Carrick Hill area of the city last June.
Both he and the driver of the vehicle have received and accepted informed warnings over the incident.
The MLA said he was trying to speak to police about the arrest of a young man just moments before he was pushed down the street on the front of the PSNI vehicle.
Tensions had been raised after the annual Tour of the North parade passed through the area.
On Thursday, Mr Kelly released a statement, part of which read: "After careful thought I have instructed my solicitor to withdraw the civil writ against the Chief Constable of the PSNI in connection to an incident in Carrick Hill in June 2013.
He claimed: "There are a number of facts about the incident which I believe I should set out: - The police Land-Rover was at a standstill when I was remonstrating with the driver
"I took hold of the grill on the bonnet as the vehicle moved off and gathered speed.
"I believe the driver's actions were both unacceptable and dangerous and this can be witnessed from the video released."
Mr Kelly revealed his plans to take legal action against the PSNI on Monday.
On Thursday night, he said he decided to drop the case because "issuing the writ seems to have increased local political tension and created an unwanted diversion from more pressing issues.
"So in this instance I have decided not to pursue the case," he added.
© UTV News