Alexander Nix said that he could not do so while an inquiry by the Information Commissioner's office (ICO) was live.
He also accused the watchdog of "abusing his parliamentary privilege" by using details revealed at an earlier appearance as the basis to raid his firm's offices.
Cambridge Analytica announced it was closing in May.
It became the centre of a data-sharing row after it emerged the firm had acquired data from Global Science Research (GSR), which in turn had collected millions of Facebook users' personal details in breach of the social network's rules.
Cambridge Analytica denied any wrongdoing before its collapse and said the disputed Facebook data had been deleted in any case.
Mr Nix said he stood by earlier testimony that GSR's Facebook data had proved "fruitless".
He also denied claims by an ex-employee, Christopher Wylie, that the information was the foundation of Cambridge Analytica's business.
Instead, Mr Nix said, the "cornerstone" had been voter registration information, lifestyle data and millions of other data points that it had acquired commercially.
Mr Nix first appeared before MPs in February as part of the Digital, Culture, Media and Sport Committee's investigation into fake news.
He was recalled to appear after chairman Damian Collins said there had been "serious inconsistencies" between the original testimony and evidence since received.
Mr Nix admitted that he had misled the committee when he had previously stated that Cambridge Analytica had not used data supplied to it by GSR, a firm set up by the University of Cambridge's Dr Alexsandr Kogan.
Mr Nix was asked whether he had previously misled the committee about GSR.
"I accept some of my answers could have been clearer," he said.
He claimed that he had believed he had been asked whether Cambridge Analytica was currently using GSR's data rather than if it had ever done so.
"At that stage we had deleted all of that data," he explained.
"But, on listening back to the transcript it was clear I was asked whether we had ever been supplied with that data and my answer should have been yes.
"At the time we had no reason to believe that it had been collected improperly."
Mr Nix blamed much of the controversy on whistle-blower Mr Wylie, who he described as "bitter and jealous".
He said that Mr Wylie - who is in attendance - had been the "single source" for most of the accusations made against Cambridge Analytica.
The committee's chairman challenged this claim.