If all space is filled with rock except a spherical cavity, what is the effect of gravity inside the cavity?

First of all, this is a very interesting question and shouldn't be
dismissed lightly.  The fact that so many wrong answers have been
posted so far shows that.  After thinking about it for a while, I found
the solution.

>Anyone who has studied physics knows that a spherical shell of matter
>produces no gravity in its interior.

Yes, that's true.

>So if such shells progress outward to infinity, the net effect should
>be zero inside the cavity.

No.  That would be a different situation.

>On the other hand, if all space is full of rock, then by symmetry
>there should be no gravity in any direction.  And a ball of rock
>produces a gravitational pull toward the center, so if we remove a
>ball of rock it should leave a gravitational pull outward: free
>objects in the cavity should fall to the "floor".

Yes, this is exactly what will happen, and no, Isaac Newton does not
disagree.  Consider this:

Suppose you are inside of a big homogeneous ball of rock with a 4000
mile radius, such as an imaginary planet.  If you are 1000 miles from
the center, you will feel a relatively weak gravitational pull toward
the center.  If you get closer, this force will weaken and will finally
reach zero at the center.  An oversimplified explanation of this force
is that 1000 miles above the center, you have more rock below you than
above you.

Now imagine that a hollow sphere with a radius of 1000 miles is somehow
created under your feet, centered at the center of the planet.  The
calculus result mentioned by others here shows that at any point within
that sphere, including the place where are standing, the net
gravitational force will be zero.  The removal of that rock creates a
situation where you have a gravitational balance between a lot of rock
far from you and a lesser amount closer (again, an oversimplification.)

Now to consider your example of space filled with rock.  Someone
mentioned that space is not really infinite because of the
four-dimensional curve of the Universe.  That doesn't necessarily have
to be true in your imagined scenario, but even if it is, it wouldn't
affect the result.  Infinite space or not, if all of space was filled
with rock, there would be no gravitational force felt anywhere.  There
would be no place where you could stand and have more rock below you
than above you.  This is what makes all of the difference.

If a hollow sphere was somehow carved out of rock-filled space, then
there would be gravitational force present everywhere inside the sphere
except the center.  This force would push away from the center,
increasing with the distance.  The force would reach a maximum at the
surface, and then continue outside of the sphere, repelling matter from
the sphere with a force that would weaken with distance.  Thus, in a
way, this would be a case where gravity would actually repel instead of
attract.

-- Mike Naylor <you.know.him@you.love.him>, http://www.serve.com/games/