Wright went off inside the opening 10 minutes in a test in which Australia also lost scrumhalf Nic White and centre Joseph Suaalii to concussions.
The defeat followed a stunning comeback for a 38-22 victory in Johannesburg the previous week, though that also took its toll with injuries to captain Harry Wilson (knee), Dylan Pietsch (broken jaw) and James Slipper (concussion).
Australia are down to the bare bones at fly-half, with Noah Lolesio, Tom Lynagh and Ben Donaldson all sidelined, leaving James O'Connor to win his first Test cap in three years in South Africa.
Scans have confirmed Wright's injury, and he will see a specialist on his return home this week. Recovery from ACL injuries can range from three to 12 months.
Australia host Argentina in two home matches in the Rugby Championship next month and have home and away clashes against New Zealand to finish the campaign.
"Our strength in depth is going to be tested," coach Joe Schmidt said after the loss in Cape Town.
"The guys who are experienced, they’re going to have to get around the new guys who come in and we’re going to have to calibrate those guys as quickly as possible, assimilate them into the group and try to kick on against Argentina in two weeks’ time.
"After today and even losing three guys last week, the attritional nature of the tour is going to make it a little bit more difficult coming up against Argentina.”