"Andrew and I are deeply sorry for this", he said.
Separately, Sydney Morning Herald and The Age reports on Thursday about suspected large-scale fraud by Mr Thorburn's former chief of staff, in which insiders described a culture of largesse and extravagant spending in his office, put his executive oversight in question.
2GB Radio host Alan Jones said this week that Baird "would be a very good CEO", while current NSW Premier and Baird's former treasurer Gladys Berejiklian called him "an outstanding individual".
Royal commissioner Kenneth Hayne's final report - made public on Monday - expressed serious concerns about the pair's leadership of the bank and failings that included charging fees for no service.
"I'm sad to be leaving NAB in these circumstances but it is absolutely the right thing to do".
The NAB board will now initiate a global search process for the CEO role while actively considering a range of internal candidates.More news: Officials say man shot at border is alive
"He's been in the industry for decades and understands the importance of aligning the customers' interests with the bank's".
"I have had a number of conversations with the Chairman this week. However, I recognise there is a desire for change".
Thorburn said he had offered to step down and the board had accepted.
"I have devoted myself to serve NAB and my hope and ambition is that the bank can move forward and achieve its vision of being trusted by customers for exceptional service".
Dr Henry had been NAB's chairman since 2015 and said he does believe the bank is in a better shape today than when he joined its ranks.
It said customer remediation programs and regulatory compliance investigations continue to add to costs, although amounts and timing were "uncertain". "It just gives NAB an opportunity to reset".
"I am enormously proud of what the bank has achieved and equally disappointed about what the royal commission has brought to light in areas where we have not met customer expectations". He is an executive of the highest quality and integrity, and I am deeply impressed that he has demonstrated accountability for the issues NAB faces.More news: Kendrick Lamar, Drake and Childish Gambino reportedly turned down the Grammys
Although Henry insisted the decision was "not made in reaction to any specific event", in a teleconference after the announcement Thorburn conceded it was those observations by Hayne that had "caused the discussions this week" leading to their resignations.
Since replacing Cameron Clyne as CEO, Mr Thorburn had overseen the divestment of the bank's unprofitable Clydesdale Bank and 80 per cent of its MLC wealth management business, and embarked upon a $1.5 billion restructure that will trim 4000 jobs over three years.
"I was not persuaded that NAB is willing to accept the necessary responsibility for deciding, for itself, what is the right thing to do, and then having its staff act accordingly". I thought it telling that Dr Henry seemed unwilling to accept any criticism of how the board had dealt with some issues.
Mr Thorburn's future has been subject to speculation since he announced surprise plans to take leave around the release of the Hayne report.
Analysts and commentators quickly concluded that Henry and Thorburn appeared to be in an untenable position.
It halted trading in its shares ahead of the announcement.More news: England's Chris Ashton gets rare Six Nations start
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