The 33-year-old made the move to St James' Park from Swansea City in 2018 and has represented the Magpies on 89 occasions in all competitions, but he has not been involved at the start of the new season.
Fernandez had not even been included in the squad for any of the team's opening five matches of the new Premier League campaign, and he has now departed Eddie Howe's side for Elche.
"A popular player with supporters, staff and teammates, he struggled with injuries last term and played only twice under Eddie Howe, but the head coach has previously spoken of the former Napoli man's influence off the pitch, while he frequently demonstrated his abilities on it during his time on Tyneside," read a statement on Newcastle's official website.
"And he has now signed for the La Liga side, who are currently 17th in Spain's top flight, on a permanent basis. Fernandez departs with the best wishes of everybody at Newcastle United and the club thanks him for his service."
Fernandez - a 32-time Argentina international - started his professional career with Estudiantes, while he also represented Napoli and Getafe ahead of a switch to Swansea in 2014.
The centre-back could make his debut for his new club against Villarreal on Sunday evening.