Messi was forced off in the 65th minute of Sunday's contest after he went to ground and immediately signalled to the Argentina dugout.
The 37-year-old took off his right boot and walked slowly off the pitch, before he was shown in tears on the bench with a visibly swollen right ankle.
Messi still managed to clinch his 45th career trophy as Lautaro Martinez scored an extra-time winner to clinch Argentina a record-breaking 16th Copa America title.
Inter Miami provide Messi injury update
With the MLS season in full swing, Inter Miami will hope to have their star man back on the pitch as soon as possible.
However, they released a statement on Tuesday confirming Messi has sustained an ankle ligament injury and would be sidelined indefinitely.
The statement read: "After a medical evaluation, it has been determined that Leo Messi has suffered a ligament injury in his right ankle. The captain's availability will be determined by periodic assessments and the progress of his recovery."
Messi has been in fine form for Inter Miami this season, having scored 14 goals and provided 11 assists in 15 competitive appearances.
Inter Miami, who currently sit in second place in the Eastern Conference, will face Toronto on Wednesday and Chicago Fire on Saturday, before they turn their focus to the defence of their Leagues Cup title.
Messi breaks silence on ankle injury
A day before Inter Miami's announcement, Messi took to social media to reassure fans that he was "fine" after his Copa America final appearance was cut short by injury.
"The Copa America is over and the first thing I want is to thank everyone for the messages and greetings," Messi said in a post on Instagram.
"I'm fine, thank God, and I hope I can soon be on the field again enjoying what I like to do most. I am happy, very happy, especially because we achieved the goal we had and Fide (Angel Di Maria) leaves us but with another Cup. The older ones, like him, Ota (Nicolas Otamendi) or me, experienced it with special emotion.
"With other teammates who have already played several tournaments and added their experience as well, and with a group of young people who give their all with every ball. We are a team and also a family, a spectacular group. Thanks to everyone who supported us, this national team has a lot of present and a lot of future too."
While Di Maria has retired from international football, Messi has indicated that he has no immediate plans to call time on his Argentina career.