Ms Driver is the first celebrity to leave the worldwide charity, which is headquartered in Oxford, UK, since an investigation by a British newspaper revealed senior Oxfam staff had paid prostitutes for sex in earthquake-hit Haiti in 2011.
Fears the scandal will hit public confidence has led to the country's global aid and development groups swiftly tightening foreign conduct policies for staff and contractors.
"I am desperately sorry for the harm and distress that this has caused to Oxfam's supporters, the wider development sector, and most of all, the vulnerable people who trusted us", she said.
He added: "The Scottish Government will not hesitate to take appropriate action whenever we suspect that any abuse has occurred, and we expect the same commitment from those working with us".
The scandal broke last week with a report in The London Times.
Oxfam's corporate sponsors are reviewing their relationship with the company and the charity is also under investigation by the United Kingdom government's Charity Commission agency.
Meanwhile the Department for International Development (DFID) said Friday it was reviewing its relationship with the UK-based charity, to which it gave almost Â£32 million (36 million euros/$44 million) past year.
The Scottish Government has awarded Oxfam Scotland more than £7 million in funding since 2008.
Pressure is now mounting on Mark Goldring, Oxfam's chief executive, who will be summoned to appear before a committee of MPs to be questioned about his response to the scandal.
Oxfam was in the midst of a large effort on the island after the quake, which killed more than 200,000 people and left many more injured and displaced.
"But I don't think it was in anyone's best interest to be describing the details of the behaviour in a way that was actually going to draw extreme attention to it".
Even in this time of #MeToo revelations, allegations of misconduct against seven former aid workers with Oxfam Great Britain are particularly shocking.
Both Oxfam and Save the Children Australia say they are urgently reviewing their staff-vetting and monitoring procedures in light of the developments.
The charity commission is investigating claims that he was able to get a job with Oxfam just two years after being dismissed from health charity Merlin.
The probe led to the sacking of four Oxfam employees, the charity said on Friday, adding that three people, including the country director, resigned before the end of the investigation.