A challenge from Saints defender Kyle Walker-Peters left Garnacho in pain, forcing United boss Erik ten Hag into withdrawing his second-half substitute. Garnacho later left Old Trafford on crutches.
The 18-year-old winger was primed for his Argentina debut after a stellar season where he has four goals and four assists in all competitions, but he will now have to wait to don the blue-and-white kit of the World Cup winners.
"It is difficult to put into words how I'm feeling right now. Unfortunately I will not be able to help my team and team mates in the upcoming games in what is a very important part of the season for us at Manchester United," he wrote on Instagram.
"I am disappointed to also miss the opportunity to be with my team mates of the Argentinian national team in what would have been a great and proud moment for me and my family.
"This is part of football and our profession, however, I am already focused in my recovery."
United enter what could be a season-defining period where they take on Real Betis in the Europa League last-16 second leg on Thursday followed by an FA Cup quarter-final against Fulham on Sunday.
Argentina face Panama and Curacao in friendly matches later this month during the international break.
The challenge on Garnacho led to Walker-Peters receiving racist abuse on social media, which prompted Southampton to get the police involved.
Walker-Peters later said it was never his "intention to injure a fellow professional", with Garnacho saying he had no hard feelings.
