David Cameron has described the outburst by Tory chief whip Andrew Mitchell as "deeply regrettable", adding that the incident "must never happen again".
The prime minister, who has been dogged by questions about the affair on his current trip to the United Nations in New York, once again made clear that he did not believe matters should go any further.
Speaking at a press conference at UN headquarters, Cameron said: "As I have said very clearly, he gave an apology, it was right he apologised, what happened was wrong, it shouldn't have happened and must never happen again.
"On the basis he has given an apology and the police have decided not to pursue that any further, that is where matters should rest."
Pressed in New York over why he appeared to accept Mitchell's account of his tirade rather than that recorded in police logs, Cameron said: "What I am saying very clearly is that what happened was wrong, it shouldn't have happened, it was deeply regrettable and that is why the Chief Whip has made a very public and clear apology.
"It was the right thing to do that.
"On the basis of that, on the basis of the fact also that the police aren't proposing to take matters further, that is how things stand."
Cameron called Mitchell in for a face-to-face meeting on Thursday, the day after his outburst, and told him his behaviour was unacceptable.
In response, it is understood the Chief Whip looked the PM in the eye and said that he did not use the words attributed to him in reports but accepted he needed to apologise.
It emerged on Tuesday night that Mitchell repeated his request to be allowed to cycle through the main gates the day after his outburst, but was again told to use the pedestrian entrance.
But it is understood that there was no repeat of the angry confrontation on that occasion.
Labour has demanded Cameron establish which account of the incident is true and accused him of a cover-up.
According to the 442-word police log - published in full in the Daily Telegraph - Mitchell called officers "f****** plebs" and told them to "learn your f****** place".
The female officer who made the report said she warned him that if he continued to swear, she would have no option but to arrest him under the Public Order Act - a stance backed by senior Tory Boris Johnson.
The London mayor, who has consistently argued that anyone who swears at police should face arrest, described the police action as "wholly common-sensical".
Among police representatives pressing for an inquiry, there appeared to be reluctant acceptance that a decision by Cabinet Secretary Sir Jeremy Heywood to rule out a formal inquiry marked the end of the road.
Irene Curtis, president-elect of the Police Superintendents' Association of England and Wales, said she was "disappointed" because, in denying the police account, Mitchell was "challenging the officer's integrity".
However she added: "As far as we are concerned it's over."
Paul McKeever, chairman of the Police Federation representing rank-and-file officers, agreed that Mitchell was questioning the officer's integrity but said the politician's future was "a matter for the Prime Minister".
Energy Secretary Ed Davey said Mitchell would regret his "disgraceful" behaviour for many years, but his apology should be the end of the matter.
He said: "I understand he has apologised to the police officer he abused, that's quite right, it was disgraceful behaviour.
"But if he has made a fulsome apology to the police officer that can draw the line under this episode."
The Liberal Democrat Cabinet minister added: "I think we should condemn people who swear at police officers. The police do an incredibly difficult job and deserve our respect and I think Mitchell will regret, for many years, the fact that he didn't respect a police officer."
On Tuesday, Boris Johnson weighed in on the chief whip affair, with the Mayor of London saying he was pleased that the police were on the verge of arresting Mitchell for swearing at them at the gates to Downing Street.