Valero wrapped up his switch to San Siro on Monday after passing a medical at his new club, the 32-year-old moving on from Fiorentina in bitter circumstances after five years with La Viola.
Fiorentina previously indicated the club is considering legal action against Valero over an audio message from the Spaniard that was released on social media.
Valero then hit back at the club and pledged to reveal "the truth" in a press conference, amid reports he felt he was being forced out of the club.
Inter are yet to confirm the signing of Valero or the length of his contract, but Fiorentina announced his departure in a statement.
"Fiorentina thanks Borja for the splendid five years spent together and wishes the Spanish midfielder the best of luck," the Florence club said.
Valero replaces Ever Banega among Inter's midfield options under new coach Luciano Spalletti after the Argentina returned to Sevilla after a single season in Milan, while Fiorentina's star winger Federico Bernardeschi has been linked with a move to Serie A champions Juventus.