Does a sling or a gun exert more initial force of the projectile?

It's kind of important to know an accurate weight for the rock you used if you want to calculate the impulse required to accelerate it. Looking up the weight of a similar small rock won't do.

Instead, let's find some figures from the internet to use. The muzzle energy of an average .22 LR pistol is about 160 Joules, according to the wikipedia entry below. This means that the bullet leaves the muzzle of the gun with 160 Joules of kinetic energy.
http://en.wikipedia.org/wiki/Muzzle_energy

In the following video, they claim that the world record kinetic energy for a handheld slingshot projectile is 113 Joules. This is less energy than a .22 bullet, and chances are you're firing them off with quite a bit less energy than the world record holder.
http://youtu.be/1v4TEX2erog

So, in short, a projectile from a handgun has more kinetic energy (and requires a greater impulse) than a projectile from a handheld slingshot.

 

More kinetic energy, yes. More impulse -- I don't think so.

The kinetic energy of a projectile scales as the square of its speed. The momentum of a projectile scales linearly with its speed. That means that a fast-travelling .22 slug has lots of kinetic energy but relatively little momentum compared to a slow-travelling sling stone.

"Impulse" is a measure of momentum, not energy.