Mr Musk stated on Twitter that the deal “cannot move forward” unless Twitter proves that less than 5% of daily users are fake or spam accounts.
Twitter has defended its figures, claiming that Mr Musk waived his right to “due diligence” in order to finalise the deal.
The squabble has cast doubt on the takeover.
Analysts believe Mr Musk is looking for ways to renegotiate the deal’s price or walk away.
Mr Musk agreed the $44bn (£34.5bn) deal to buy Twitter with its board in April, but last week Mr Musk said the deal was “on hold” while he sought details about fake accounts.
In tweets and public appearances, he has continued to press the issue, implying that he may seek a lower price.
Mr Musk said on Tuesday that Twitter CEO Parag Agrawal had “publicly refused to show proof” that fewer than 5% of its accounts were fake, and that the deal “cannot move forward” until Mr Agrawal did.