It’s Newton’s Third Law/Conservation of Momentum. If something pushes something away, then that thing is pushed away just as much. Ballistic weapons have recoil because the expanding gas pushes as hard on the gun as on the bullet, just you’re holding the thing so most of the actual energy is imparted on the bullet.
A centrifugal gun is going to have the same issue, the centrifugal force pushing the bullet out is going to impart a centripetal force of equal value on the gun. It would probably be fairly small since it’s just a small projectile at a relatively low speed, but it would certainly exist. Especially at 20 rounds/second or whatever you’re expecting to get out of this.