Polls have shown declining support for Mr. Rajoy and his party, though the next general election could be more than two years away. The party held a majority in Parliament from 2011 to 2015, but elections in 2015 and 2016 gave no party anything close to a majority, and Mr. Rajoy struggled for months to form a government.

He has also been engaged in a struggle with separatists in Catalonia, taking an approach that critics contend has prolonged the political deadlock. After trying unsuccessfully to suppress an independence vote last year, he dissolved the Catalan Parliament and called new elections, only to see separatists again win a narrow majority.

Ms. Cifuentes is expected to stay on as a regional lawmaker, ahead of regional and municipal elections across Spain in 2019.

The case of Ms. Cifuentes is “exactly the opposite of the exemplary attitude that one expects from somebody who intends to be a true leader,” wrote Francesc-Marc Álvaro in a recent column for the newspaper La Vanguardia. If a politician was found to have a bogus degree, he added, “in other countries like Germany and Britain, we know how these things end: with the immediate resignation of the person who deceives.”

Prosecutors are now investigating the broader practices of King Juan Carlos University, which had been attended by other politicians from Spain’s governing party. There are also concerns about financial mismanagement at the institution, known in Spanish as Rey Juan Carlos University, which was hit by a major plagiarism scandal in 2017.

On Wednesday, Ms. Cifuentes acknowledged for the first time that she received her degree as a result of “agreements with the professors,” but she did not elaborate. When Ms. Cifuentes had defended herself over the past month, some university professors said that their signatures had been forged on documents relating to her degree.

The case of Ms. Cifuentes “is doing a lot of damage not only to the Rey Juan Carlos but to all the universities of this country,” Miguel Sebastián, a former Socialist minister and university professor, recently told La Sexta, a television channel. He called it an affront to students who worked hard to get their degrees and their families who had supported them.