As can be seen in this video, a CNC machine uses a cutting wheel in order to cut through fabric. That tool resembles a pizza cutter.
So can a pizza cutter cut through fabric? If so, what kind of fabrics is it able to cut?
There's a reason people keep separate fabric scissors from their regular scissors--cutting through fabric requires a very sharp blade, which you're unlikely to find on the average pizza cutter (designed to cut through soft cooked materials).
What you're seeing is a rotary cutter, a circular blade on a handle that's sharp enough to cut through fabric. Unlike a typical pizza cutter, rotary cutters come with blade guards and retractable blades to protect the user from cutting themselves. They're frequently used by quilters, who need to cut many pieces with straight lines, often using heavy plastic templates. In the images shown on the blog Quilter's Review's review of several cutters, you can see the safety features on the handheld rotary cutters. The CNC machine is using a similar blade, but as it's not being handled by a human directly, the need for the additional safety features is reduced.
Yes, if it is sufficiently sharpened. But... no restaurant would ever do that. I've worked in multiple restaurants including Pizza Hut (wow, that was 30 years ago).
Most places used a rotary pizza cutter, except Pizza Hut had long curved blade at the cut station for normal pizzas. We called it the bat'leth*.
Every utensil was made of high quality stainless steel that could have been sharpened to the point that you could shave with it, but there was no need. A moderately sharp edge is enough to cut cooked pizza dough with that satisfying crunch. Anything more would just be asking for a ride in an ambulance and a stack of demolished cutting boards.
*Once you got the rhythm down, rocking that blade across a large pizza 6 times to get 12 slices felt like Klingon battle training. Granted I was also a teenaged Star Trek nerd.
Just for fun, I tried it, with a well-used pizza cutter and scrap materials on a cutting mat. Though the pizza cutter is old, it still does its intended job of cutting pizza very well.
It wouldn't go through thin cotton (probably an old pillowcase, that I use for checking the sewing machine is running properly). It wouldn't go through light ripstop polyester, and it wouldn't go through flat elastic.
I pressed rather hard, as well. It would be better for producing crease lines than actually cutting (though there's probably some damage to the fabric)
It depends on the quality of the knife.
As mentioned in a comment, a pizza knife is a circular knife (also called a 'roller blade'). Like any knife it can be high or low quality, sharp or relatively blunt, strong or weak, which will be decisive for the fabrics you are able to cut with it.
If you have a specific kind of fabric in mind, my suggestion would be to simply try it out, but, as Allison C points out, it's better to get a rotary cutter.
The advantage you might notice is that a rolling blade allows for easier maneuvering, and can be especially useful - if sharp and sturdy enough - to cut out drawn patterns.
As suggested by Elmy below, be sure to use a proper support for cutting. A self-healing cutting mat is a very good option for this.