The Blaugrana youngster, who spent last season on loan at Mallorca, has signed a three-year deal with the Galicians and will join them for 1 million euros.
The defender made eight Liga appearances for the Blaugrana, spending most of his time at the club with the youth ranks. An anterior cruciate ligament injury suffered against Osasuna in January 2012 hampered his progress, and he spent last season on loan at Mallorca.
Fontas revealed that the chance to link up with former Barca B manager Luis Enrique was key to his decision to join the Galicians.
"[I've chosen Celta because] of the club's project, the way they play and because of the spirit Luis Enrique will pass on," Fontas told the club's official website. "Celta chose me and that made me decide to come and sign a three-year contract."
The 23-year-old was used primarily as a defender during his time at Barca but says he could be deployed in a midfield role at Balaidos.
"I'll be in the position in which the coach puts me," Fontas said. "I've played more at center back than in midfield but I also see myself as a defensive midfielder.
"The important thing is to perform at my best."