Can I apply the stabilization (Warp Stabilizer) to another layer?

It sounds like you are trying to motion-track part of your footage and add something to the shot. If that is what you are trying to do, your workflow is backward. 

Here is how I apply Warp Stabilize to a shot that must include motion-tracked layers:

1. Motion track the object or objects in the shot that I want to use for the composite
2. Apply motion tracking to the new layers so that they follow the objects in the original footage
3. Apply the final touches to the composite to polish the shot
4. Pre-compose the original footage and all layers used in the composite
5. Run Warp Stabilize on the pre-comp (new nested composition)

That is the only way to apply the same warping stabilization to other layers you add to the scene. I always create my composite before running Warp Stabilize on any shot that needs to have that tool applied to the shot. It is the only way to keep the new layers lined up.

Could you mask you footage, then pre-comp it and run the Warp Stabiliser on the pre-comp.  Once you have it the way you want, you should be able to remove the masks and have the footage stabilised... but that's a guess.  As Mylenium says, if you can tell us more and share a screenshot of your footage, other workflows might be a better fit.

Long and short: Nope. The Warp Stabilizer does not expose its internal vector synthesis, so there is no way to transfer that data outside of the black box that the effect is. Since you haven't provided any details on what you are exactly trying to do we can't advise on other techniques/ worklfows that may solve your dilemma.

Mylenium