Sir Tom McKillop, who will retire next year, said he is "profoundly sorry" for the bank's losses and its current situation, describing it as the worst experience of his 40-year career.
Under the bailout deal, which was backed by 99 per cent of shareholders, RBS will attempt to raise Â£15 billion by issuing shares at 65.5 pence each. If any of these shares remain unsubscribed, the government will buy them.
The state will also purchase around Â£5 billion worth of preference stock directly from RBS.
Overall, it could end up with a 60 per cent stake in the 281-year-old bank.
"The challenges we must now address as an institution, as a country and indeed as part of the world's financial system, are unprecedented," Sir Tom said.
In other news, shareholders in Lloyds TSB have approved the bank's proposed takeover of the country's largest mortgage lender, the HBOS group.