Officially known as "Dirty Corner," the giant steel funnel that Kapoor himself has described as "very sexual" was covered in anti-Semitic graffiti, said Versailles president Catherine Pegard.
Phrases such as "Queen sacrificed, twice insulted", "the second rape of the nation by deviant Jewish activism" and "Christ is king in Versailles" were daubed in white paint on the sculpture by the British-Indian artist.
President Francois Hollande condemned the vandalism as "hateful and anti-Semitic" in a statement and Prime Minister Manuel Valls took to Twitter to express his "disgust".
France's Culture Minister Fleur Pellerin said the act of vandalism was an "attack on freedom of creation" and condemned the "unacceptable" phrases painted on the funnel.
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"This is no more or less an act which reveals a fascist vision of art," she said in a statement after going to Versailles to inspect the damage.
The steel-and-rock sculpture, which faces the royal chateau, is 60 metres long and 10 metres high, and attracts five million tourists a year.
"This act of intolerable violence against the work of an international artist shocks and saddens me," Pegard told journalists.
This time, the tapered funnel was sprayed with white slogans, as were the rocks surrounding it.
A source close to the investigation said the "subliminal message was that France has lost its shine of days gone by compared to other nations. A message with royalist leanings."
"We know individuals with ultra-conservative leanings, we have some ideas about those who fit the profile," the source added.
The sculpture is one of several by Kapoor which are on show in the gardens and inside one room of the palace until November.
"The point is to create a dialogue between these great gardens and the sculptures," he said before the display was unveiled.
Kapoor's exhibition is one of the most controversial at Versailles since the authorities opened the palace and its grounds to contemporary artists, hosting works by American artist Jeff Koons in 2008 and by Japanese artist Takashi Murakami in 2010.
"The very controlled landscape of Versailles is drawn into instability.