To stabilize a projectile there's a minimum rotational speed that must be achieved.
Given typical rifling twist rates, a bullet will accomplish maybe one full turn within the barrel, possibly two, but the rotational speed imparted by the combined vectors of forward motion at several hundred fps, or a few thousand, depending on the cartridge and barrel, is enough to spin the bullet at hundreds or thousands of rpm, and thus impart reasonable gyroscopic stability to the projectile.
So you have a number of factors necessary to achieve stability:
- The projectile must maintain a steady rotational axis during the acceleration period;
- the projectile must rotate at some minimum speed to be stable in flight;
- the projectile must achieve enough velocity in flight that it's worth the trouble of stabilizing it.
In a firearm or airgun,
- is achieved by a rigid barrel, along which acceleration is performed;
- is achieved by integrating rifling into the acceleration tube;
- is achieved by the application of sufficient pressure in a short time frame.
I'm thinking that, using a slingshot, it's going to be pretty damned hard to get all of that to happen.