The First Minister refused to say why Eilidh Mactaggart, who was appointed chief executive in 2020 on a salary of £235,000 a year, quit her role last week.
She said ministers were notified of the move earlier in February, but details of the resignation were “confidential”.
Ms Sturgeon’s spokesman later stressed it was a “personal employment issue”.
MSPs are now set to summon SNIB chair Willie Watt to Holyrood’s finance and administration committee in a bid to get answers.
Speaking during First Minister’s Questions on Thursday, Scottish Conservative leader Douglas Ross accused Ms Sturgeon of “secrecy” and “shutting down scrutiny” over the issue.
Ms Sturgeon said: “Everyone across the chamber will understand that I’m not going to go into the confidential details of anybody’s employment situation here in the chamber.”
She added: “This issue is not a matter for Scottish Government ministers, it’s a matter for the board of the Scottish National Investment Bank. Ministers had no input into that, although we were told earlier in February that the chief executive would be leaving the bank imminently.”
Ms Sturgeon insisted the bank, which was launched in November 2020, was performing "exceptionally well”.
The public body makes investments in Scottish firms and projects, with ministers pledging funding of £2 billion over its first ten years.
Mr Ross said: “The First Minister has used this opportunity to explain how well the bank is doing and the vital work they are undertaking and therefore I think it is important that this Parliament knows and the public in Scotland know why the chief executive resigned so abruptly earlier this week.”
The Scottish Tory leader said Ms Mactaggart had resigned “just days before” the launch of the SNP’s new ten-year economic strategy, calling the timing “suspicious”.
Ms Sturgeon said this was a coincidence, adding: “On the issue of the former chief executive – the former chief executive of the Scottish National Investment Bank is a private individual.
“She has opted to resign her post as chief executive of the bank. She is entitled to the duty of care and the confidentiality that any other individual in her circumstance is entitled to.
“It would be completely wrong – and I think most reasonable people would accept – that it would be completely wrong of me in the chamber of the Scottish Parliament to breach that confidentiality.”
Speaking after FMQs, Mr Ross said: “This is all starting to look very murky. Nicola Sturgeon completely dodged every question about the abrupt resignation of the chief executive of the Scottish National Investment Bank.
“The ministerial code demands that information cannot be hidden from the public unless there are ‘clear and lawful reasons’ for doing so. We’ve no idea if that’s the case because Nicola Sturgeon won’t say a word about it.
“This secrecy and shutting down scrutiny is completely unacceptable, especially when this bank will eventually be in charge of £2bn of public money.”
A spokesman for the First Minister later confirmed she was aware in “very broad terms” of the reason why Ms Mactaggart resigned.
A spokeswoman for SNIB would not comment further, but confirmed no non-disclosure agreement was in place.