The 27-year-old suffered the injury to his right hand during Saturday's 45-20 defeat against Australia and will now need an operation to repair the damage.
Moyano's injury came as the Pumas crashed to their fourth successive defeat in the Rugby Championship to leave them rooted to the foot of the table.
Afterwards, head coach Daniel Hourcade said: "We had a good first half and a bad second.
"We failed to score tries and they did not miss anything. Australia ended up being a clearly dominant. When they scored against us, we lost our heads and that was the difference.
"To have opportunities, you must maintain your level for the whole game. The Australians took every opportunity. We did not."