Jones had to be replaced in the eighth minute of the warm-up Test at Murrayfield on Sunday (AEST) after sustaining the injury while being cleared out of a ruck as he attempted to get his hands on the ball.
Lions head coach Gatland confirmed after the match that Jones will not fly out to face the world champion.
The New Zealander said: "He's dislocated his shoulder.
"Prognosis of him being fit and very, very best-case scenario is the first Test [in Cape Town on 25 July AEST]. He's going to go and get a scan but we’re looking like we’re going to have to replace him."
Gatland added: "It’s pretty much made [the decision that Jones will not travel to South Africa], I was speaking to the physios, I’ve spoken to Alun Wyn.
"He understands, very best case scenario, he may get back for the Test but we need to call someone in and hopefully get them on the flight tomorrow.
"It's very disappointing, he’s been around rugby long enough, he's going to be gutted, I'm very disappointed for him. It's one of those things, you’ve got to put that behind you and move on as quickly as possible you can."
Gatland will consider who to replace lock Jones with in the squad and a new skipper.
He added: "We are discussing a couple of names at the moment, unfortunately I got called away to do a press conference.
"I'm going back to finalise that after I've spoken to you guys, we’ll make that call, have a discussion about who comes in and potentially who’s going to take over as tour captain as well."
Justin Tipuric also suffered a shoulder injury early on but Gatland is unsure of the extent of the damage.
The former Wales boss said: "It was a sting in the shoulder. We're going to check, to make sure there's no nerve damage.
"If there was nerve impingement that could rule him out - at the moment, pretty sure he’s going to be OK."
A hamstring injury sustained by Jack Conan will also be checked.