His deputy Jonathan Williams has also agreed to voluntarily step back from normal duties “with immediate effect” until the independent review into thefts has concluded.
In a statement, Fischer said it was evident the museum “did not respond as comprehensively as it should have” when it was told about the thefts in 2021.
He also withdrew remarks he made earlier this week about the art dealer who first alerted museum bosses.
Fischer said he expressed “sincere regret” over the “misjudged” comments.
The Metropolitan Police is investigating items that are “missing, stolen or damaged” from its collection.
An unnamed member of staff - since interviewed by police - has been sacked and the institution is taking legal action.
It is understood that the items, which include gold jewellery, gems of semi-precious stones and glass, were taken before 2023 and over a “significant” period of time.
Leaked emails claim the museum was alerted by an antiquities dealer to items being sold on eBay in 2021 and ignored the report.
German art historian Fischer, who the London institution announced in July will step down from his role next year, left on Friday afternoon.
He said in a statement: “Over the last few days I have been reviewing in detail the events around the thefts from the British Museum and the investigation into them. It is evident that the British Museum did not respond as comprehensively as it should have in response to the warnings in 2021, and to the problem that has now fully emerged.
“The responsibility for that failure must ultimately rest with the director. I also misjudged the remarks I made earlier this week about Dr Gradel. I wish to express my sincere regret and withdraw those remarks.
“I have offered my resignation to the chairman of the trustees, and will step down as soon as the board have established an interim leadership arrangement. This will remain in place until a new director is chosen.
“The situation facing the museum is of the utmost seriousness. I sincerely believe it will come through this moment and emerge stronger, but sadly I have come to the conclusion that my presence is proving a distraction.
“That is the last thing I would want. Over the last seven years I have been privileged to work with some of the most talented and dedicated public servants. The British Museum is an amazing institution, and it has been the honour of my life to lead it.”
The museum has not specified how many items have been stolen or detailed what the missing items are, saying only that they are “small pieces” dating from the “15th century BC to the 19th century AD”.
An independent review of security, conducted by former museum trustee Sir Nigel Boardman and Lucy D’Orsi, Chief Constable of British Transport Police, has been launched.
George Osborne, chair of trustees, added: “The Board of Trustees has accepted the resignation of Hartwig Fisher as Director.
“He has acted honourably in confronting the mistakes that have been made. No one has ever doubted Hartwig’s integrity, his dedication to his job, or his love for the Museum.
“Hartwig had already announced his intention to step down some weeks ago, so the process of finding a new permanent Director is already underway.
“The Trustees will now establish an interim arrangement, ensuring that the Museum has the necessary leadership to take it through this turbulent period as we learn the lessons of what went wrong, and use them to develop plans for a strong future.
“The Trustees also wish to thank the many, many staff who work so hard for the Museum and keep it running.
“I am clear about this: we are going to fix what has gone wrong. The museum has a mission that lasts across generations. We will learn, restore confidence and deserve to be admired once again.”
Online marketplace eBay said: “Our dedicated law enforcement liaison team is in close contact with the Metropolitan Police and is supporting the investigation into this case.
“eBay does not tolerate the sale of stolen property. If we identify that a listing on our site is stolen, we immediately remove it and work with law enforcement to support investigations and keep our site safe.”
Tim Loughton, the chairman of the all-party parliamentary group on the British Museum, said he was sad to hear of Fischer’s resignation.
The MP said Fischer had been a “really innovative and excellent director” over the last seven years.
“This news in the last couple of weeks has obviously been very damaging and it’s been embarrassing, and it’s been taken very seriously by the director of all the trustees of the British Museum,” Loughton told Times Radio.
“But they’re also acting with one hand behind their back because the police have specifically asked the museum not to give out more information whilst there’s a police investigation going on.
“So Hartwig hasn’t really been able to give his side of the story or more details, but he’s decided to clearly do what he thinks is the honourable thing and to offer his resignation today ahead of when he was due to be standing down next year anyway.”