The online editions of Spain's leading daily newspapers El País and El Mundo featured the low-quality video of Princess Cristina from behind as she answered a judge's questions on Saturday.
The video, lasting five minutes and 23 seconds, was posted on a video-sharing website, wouzee, by a user identified only as "infanta" (princess).
The leaked video is the only content posted under that user name.
Judge Jose Castro had banned recordings at the closed-door hearing in his court in Palma de Mallorca, prohibiting anyone from entering the courtroom with a camera or telephone that could record images.
He ordered an investigation after El Mundo published a photo grab from the video on its front page on Sunday.
On Monday, the video itself was posted online and picked up by the press, prompting condemnation by the chief judge of the Balearic islands, Francisco Martinez Espinosa.
"It is being investigated and if the person is found who did it they should pay the legal consequences," Espinosa told reporters.
Cristina, 48, is at the centre of a scandal over alleged fraudulent business dealings by her husband, former Olympic handball player Inaki Urdangarin.
She was grilled for over six hours Saturday over allegations she was complicit in tax-dodging and money laundering.
As the first such hearing for a direct member of the Spanish royal family, the case has taken a damaging toll on the monarchy's reputation.
The princess, who holds a master's degree from New York University, told the judge she had no knowledge of her husband's business affairs and repeatedly replied to questions be saying she did not know or could not recall, according to lawyers who were present at the hearing.
In the video broadcast on the El Mundo site, Castro asks the princess how many foundations and non-profit associations she belonged to before she married Urdangarin, to which she replies: "I don't remember."
She was also asked whether she was aware that she was the legal owner for money deposited in a company that she jointly owned with her husband. "I am unaware of the mechanics," she told the court.