The cabinet minister said Clarkson's apologetic explanation that he had tried not to utter the n-word should be enough to draw a line under the controversy.
But David Cameron's official spokesman said the Prime Minister believed his friend's fate was a matter for the broadcaster to decide.
Clarkson became embroiled in a racism row following claims that he used the n-word while reciting the nursery rhyme Eeny, Meeny, Miny Moe during shooting for an episode of the BBC2 programme.
In the footage Clarkson is using the nursery rhyme to compare two sports cars. He said he "mumbled where the offensive word would normally occur" in two takes, and used the word "teacher" in its place in a third.
Last night he posted a video in which he said he "did everything in my power to not use that word" and was now "begging your forgiveness for the fact that obviously my efforts weren't quite good enough".
Asked if he backed calls for Mr Clarkson, well known for courting such controversy, to be axed over the latest episode, Mr Gove said: "No, I don't.
"The word in question is horrendous and shouldn't be used but I have read Jeremy Clarkson's account in the papers today, his explanation, and it seems to me that this was a word that he never intended to utter, never intended to broadcast.
"He has been clear in his apology and I think we should leave matters there.
"But it is really important that all of us should appreciate that the use of this type of language is unacceptable and it is right that anyone who uses language like this, even in error, should apologise for doing so."
Asked if the PM shared his cabinet colleague's opinion, Mr Cameron's official spokesman said: "He does share (Mr Gove's) view: it is absolutely right that there has been an apology."
Pressed on whether he also shared the view he should keep his job, he replied: "His view is that in terms of actions and the like, that's for the BBC."
As part of a series of apologies, Clarkson said: "I wish to God that my attempts to cover up that word were better than they were.
"I was simply mumbling - saying 'ner ner' or something similar, anything but the n-word. It was my mistake and I apologise for not covering it up. But if you look at the footage you can see what I'm trying to do."
In the video post on social media, a solemn-looking Clarkson said: "I was mortified by this, horrified. It is a word I loathe."
The BBC said it had "left him in no doubt about how seriously we view this".
The allegations were reported in yesterday's Daily Mirror, which claimed the footage was studied by audio forensic experts who told them the star can be heard mumbling the n-word.
Clarkson had earlier tweeted: "I did not use the n-word. Never use it. The Mirror has gone way too far this time."
The claims come days after the motoring show's producer apologised for broadcasting a "light-hearted" joke by Clarkson that sparked a complaint of racism.
An episode, filmed in Burma and Thailand and shown in March, featured a scene in which the presenters built a bridge over the River Kwai, and as an Asian man walked over it Clarkson said: "That is a proud moment, but there's a slope on it."
Somi Guha, an actress who complained to the BBC, said the use of the word "slope" was an example of "casual racism" and "gross misconduct".
In recent years Clarkson has been cleared of breaching the broadcasting code by watchdog Ofcom after comparing a Japanese car to people with growths on their faces.
He previously faced a storm of protest from mental health charities after he branded people who throw themselves under trains as ''selfish'' and was forced to apologise for telling BBC1's The One Show that striking workers should be shot.
The motoring show has also faced complaints from Indian and Mexican politicians over remarks made about their countries while filming on location.