Tony Hall said he had "not taken this decision lightly" and recognised it would "divide opinion".
However, he added "a line has been crossed" and he "cannot condone what has happened on this occasion".
Clarkson was suspended on 10 March, following what was called a "fracas" with Top Gear producer Oisin Tymon.
The row, which took place in a Yorkshire hotel, was said to have occurred because no hot food was provided following a day's filming.
An internal investigation began last week, led by Ken MacQuarrie, the director of BBC Scotland.
It found that Mr Tymon took himself to hospital after he was subject to an "unprovoked physical and verbal attack".
"During the physical attack Oisin Tymon was struck, resulting in swelling and bleeding to his lip."
It lasted "around 30 seconds and was halted by the intervention of a witness," Mr MacQuarrie noted in his report.
"The verbal abuse was sustained over a longer period" and "contained the strongest expletives and threats to sack" Mr Tymon, who believed he had lost his job.
Mr Tymon did not file a formal complaint and it is understood Clarkson reported himself to BBC bosses following the incident.
After that, the BBC's director of television, Danny Cohen, felt he had no choice but to suspend the presenter pending an investigation.
The decision caused an outpouring of support from Top Gear fans, with more than a million people signing an online petition to reinstate him.
Announcing his decision, Lord Hall said Clarkson's dismissal was unavoidable.
"For me a line has been crossed. There cannot be one rule for one and one rule for another dictated by either rank, or public relations and commercial considerations."
However, he added: "This decision should in no way detract from the extraordinary contribution that Jeremy Clarkson has made to the BBC. I have always personally been a great fan of his work and Top Gear."
In a statement, Mr Tymon thanked the BBC for a "thorough and swift investigation into this very regrettable incident".
"I've worked on Top Gear for almost a decade, a programme I love," he continued.
"Over that time Jeremy and I had a positive and successful working relationship, making some landmark projects together. He is a unique talent and I am well aware that many will be sorry his involvement in the show should end in this way."
North Yorkshire police have asked to see the BBC's internal report, saying it will be "assessed appropriately and action will be taken... where necessary".
Responding to the news, Prime Minister David Cameron said he believed that "if you do something wrong at work there can be consequences" and that "aggressive and abusive behaviour is not acceptable in the workplace".
Jeremy Clarkson took a slightly dull and failing car programme and turned it in to the biggest factual TV show in the world.
But this sacking has nothing to do with style, opinions, popularity - or even his language on the show.
It's about what stars are allowed to get away with off screen, a topic that's been top of the agenda for the BBC in recent months.
The corporation has had to overhaul all of its policies and attitudes towards bullying and harassment, and a long verbal tirade and a physical assault would have crossed the line for any member of staff.
Clarkson may be popular with the audience, and the BBC really did not want to lose him, but this was a star who admitted he was on his final warning and a corporation that was under intense scrutiny over what its top talent can and cannot get away with.
Top Gear, which is one of BBC Two's most popular programmes, will continue without Clarkson, who will now become the subject of a bidding war by other broadcasters.
The magazine show is one of the BBC's biggest properties, with overseas sales worth an estimated Â£50m a year for the corporation's commercial arm, BBC Worldwide.
350 million
Top Gear's estimated worldwide audience
1977 Top Gear began as a local show on BBC Midlands
170 plus episodes in its current format (since 2002)
3 million YouTube subscribers
1.7 million global circulation of Top Gear magazine
Whether Clarkson's co-presenters James May and Richard Hammond will remain on the show has yet to be confirmed.
All three had their contracts up for renewal this year, with Clarkson's due to expire at the end of March.
Hammond tweeted: "Gutted at such a sad end to an era. We're all three of us idiots in our different ways but it's been an incredible ride together."
May also updated his Twitter profile to say: "Former TV presenter".
Lord Hall said he had asked BBC Two controller Kim Shillinglaw to handle "big challenge" of renewing Top Gear for 2016, and to investigate how the channel could broadcast the last three episodes of the current series, which were pulled when Clarkson was suspended.
Meanwhile, Radio 2 DJ Chris Evans, has rubbished press speculation that he was to join the show.
"Not only is it not true, it's absolute nonsense," he told his listeners on Wednesday morning.
Jeremy Clarkson's contract will not be renewed after an "unprovoked physical attack" on a Top Gear producer, the BBC's director general has confirmed.