The incident happened in the mainly nationalist Carrick Hill area of north Belfast following the loyalist Tour of the North parade on Friday evening.
Mr Kelly says he was trying to stop the police vehicle as it drove off with a young man who had been arrested. The Sinn Féin MLA was filmed being carried on the front of it for a short distance.
The incident, which also saw Sinn Féin minister Carál Ní Chuilín require hospital treatment on her arm, has been referred to the Police Ombudsman.
On Monday, DUP leader and First Minister Peter Robinson accused Mr Kelly of being reckless and ill-advised in his actions.
Mr Kelly had said the incident had overshadowed alleged breaches of restrictions on the march.
"I don't like the way police handled the situation and let me make this clear, I have not made a criticism of all the police on the night," he explained.
"I think there was a saturation approach to Carrick Hill which I do object to, but outside of that, we are talking about an incident involving a small number of police, and it was bad policing."
On Wednesday North Belfast MP Nigel Dodds lead a DUP delegation, including Nelson McCausland MLA and William Humphrey MLA to discuss events at Friday's parade with PSNI at Tennent Street Police Station.
People in the unionist community have been pursued by police for far less and it’s important that a message goes out that nobody is above the law and that people who obstruct the police in execution of duties will be pursued.
Nigel Dodds, DUP Deputy Leader
Mr Dodds said the meeting with PSNI North & West Belfast District Commander Chief Supt George Clarke had been "thorough".
"We established a number of facts, firstly contrary to the reports from some republicans in the Carrick Hill area, they had no reports of any attacks on residents' homes last Friday night at anytime during the Tour of the North parade or thereafter," the North Belfast MP explained.
"Secondly, we established very clearly that there is a thorough investigation underway into the actions of Gerry Kelly and others in terms of obstruction of police on the night of the Tour of the North and we're very keen to ensure that the law should be seen to apply to everybody whether they're a member of the Policing Board, an MLA or an ordinary member of the public.
When questioned about potential breaches by the Orange Order bands taking part at Friday's parade, DSD Minister Nelson McCausland said the bands had adhered to the determination passing St Patrick's Catholic Church on Donegall Street.
"It was a very successful and peaceful evening and that's something that should be welcomed," he said.
When pushed to comment on the reported breach of the Parades Commission determination in the nationalist Carrick Hill area, Mr Dodds said it should be seen in perspective.
"When people are actually obstructing police, clambering onto police land rovers and inciting others and provoking others to do the same, that's a very different category of some technical breach of a Parades Commisssion (ruling)," he said.
He added that police will investigate "everything they have to investigate".
"I think when people stand back and look at the approach here, they will see a dignified, peaceful, calm, sensible approach on the part of the Loyal Orders and indeed the supporters on the evening, contrast that with the loutish and disgraceful behaviour of the Sinn Féin representative and people will make up their own minds about this."
He said with the tough stance on flag protestors taken by police and prosecutors, that Mr Kelly should not "escape the full rigour of the law" because of his position.