ACL = anterior cruciate ligament.

From the American Academy of Orthopedic Surgeons, ACL Reconstruction page:
Ligaments are tough, non-stretchable fibers that hold your bones together. The cruciate ligaments in your knee joints crisscross to give you stability on your feet. People often tear the ACL by changing direction rapidly, slowing down from running or landing from a jump. Young people (age 15-25) who participate in basketball and other sports that require pivoting are especially vulnerable. You might hear a popping noise when your ACL tears. Your knee gives out and soon begins to hurt and swell.

A bit more detail on one of their other pages ACL Injury: Should it be fixed?

Please don't take this as professional medical advice on my part, but I do recall in college that those that had ACL surgery as part of their rehab were put on a stationary bike. It was one of the first activities they did. At first it was done slowly, to test their range of motion and it was done before any other weight bearing activities. In fact it was often done before they even started walking.

Just my opinion, but I can't see how riding fixed can lead to ACL injuries. If people want to claim that it leads to other knee damage, it is tougher to argue with, but ACL injury, I just don't see it.