Riding a fixed gear bike doesn't injure people's knees.

However, an improperly-fitting fixed gear bike will reveal body issues, and most often, knee issues.

A properly-fitted fixed gear bike will actually heal knees.

A professional fitting costs money, and most bike shops only pretend they know how to do it.

Some general principles:

1. Most riders have their saddle too high: with the pedal at its bottom position, the rider should have his foot flat, or even heel down, and still have a slight bend in the knee.

2. Ride clipless, and place the cleats under the ball of the foot/big-toe, with the cleat centered between the big-toe and the "index-toe;" and, if the rider walks with his toes out, his cleats should accommodate that toe-out, a little (not a lot), so that for a toe-out walk, the cleats should point in, a little tiny bit (emphasis on "a little tiny bit").

3. Adjust the saddle fore and aft, so that with the cranks horizontal, a string hanging from the hollow just to the inside (towards the center, towards the other knee) of the knee cap passes through the spindle of the forward pedal.

4. If the rider does not race, but rather rides on the street, adjust the stem and handlebars for a more conservative, upright position: hold a pencil in each fist with the point up; sit in a hard chair with feet flat and together; extend the hands forward, as if riding; lean further forward and start to lift the buttocks off the chair; at the moment the buttocks break contact with the chair, look at the angle of the pencils and the angle of the torso; this corresponds to an all around riding position for that particular rider.

To explain the above: we have a very complex set of muscles in our hips and fronts of thighs; one of the quadriceps, the rectus femoris, starts in the pelvis and ends, sort of, at the front of the shin, without ever touching the thigh bone (femur) itself.

The rectus femoris has at least two functions, depending on the relative positions of the thigh, or femur, and the pelvis and torso.
When in a fairly upright body position, the rectus femoris can bring the thigh forward, or up (pulling up on the pedal in the back part of its spin); and then it can transition into straightening the knee (pushing the pedal over the top of the spin and mashing the pedal down in the front part of the spin).

In the racer's aerodynamic position, with the back level with the top tube and the upper body's weight on the handlebars, the rectus femoris will not, can not, pull the thigh forward, nor, then, pull up on the pedal in the back part of the spin, nor push the pedal over the top.

This means all of the quadriceps' energy goes into mashing.
Unhappily, when the quads mash, they also pull the top of the shin bone up into the bottom of the thigh bone, thus multiplying the forces in the knee joint; so that, not only do the thigh muscles push the thigh bone down into the knee joint, they also pull the shin up into the knee.

Riding a fixed gear bike puts much less stress on our knees than does running or going up stairs.
If we work on the fit of the bike, and learn how to distribute the work our legs do throughout the whole circle of the spin, our knees will get healthier and healthier.

Riding a fixed gear bike heals knees.

I expect a lot of disagreement, and I have some errands to run.

I'll check back this afternoon.