Stephen Hester told Sky News City editor Mark Kleinman: "This is about helping our customers through this, fixing the problem and making sure that what we are really here for - our customers - are serviced well.
"Our customers are expecting that, they then expect us to be accountable and so on after that and we will be."
Mr Hester also denied that the software meltdown, which has affected some 16.9 million account holders at RBS, NatWest and Ulster Bank, may have been exacerbated by a lack of IT investment.
Some critics said the group had cut technology and increased foreign outsourcing since being bailed out by British taxpayers.
"I have no evidence of that. The area, if you like, the UK backbone, has received substantial investment," Mr Hester said.
"But things go wrong. Things go wrong in technology. We have to learn the lessons from what went wrong here and try to make them less likely in the future."
The RBS boss said although some minor problems remain for Ulster Bank, both RBS and NatWest have returned to a stronger IT footing.
Although RBS has promised to compensate customers and ensure their credit ratings are not adversely affected, some observers say the next big battle for the banking group is its PR image.
"I think we are well on the road to recovery, and fingers crossed that all the bugs have been got out, but we feel that the corner has been turned," Mr Hester said.
Asked by Kleinman about his position as chief executive in the wake of the banking group's problem, Mr Hester added: "I think the most important thing I or anyone else can do at RBS is to put this thing right.
"There is plenty of time afterwards to say why did this happen, how do we stop this happening, and what's the accountability."