Will a body as big as the moon collapse inwards if it was partially hollow? Take a hypothetical scenario where the moon is hollow upto R=R/2
So will the moon collapse inwards as there is nothing to balance the gravitational force from outside?
 A: The thick hollow moon is an interesting case. On the outside it will have mass $M=(4\pi/3)(7/8)R^3\rho$ and corresponding surface gravity, but inside the shell there is nothing below, so the gravitational force (by Newton's shell theorem) is zero. That may look like the shell should be stable. 
However, every part of the inner surface will be subject to the weight of all the matter above. The total downward pressure is $$P=\int_{R/2}^R \frac{r^2}{(R/2)^2}\frac{GM(r)}{r^2} dr$$ where the first factor is due to the outer shells having greater area than the inner surface. Expanding and simplifying
$$P=\frac{24\pi G \rho}{3R^2}\int_{R/2}^R r^3-\frac{R^3}{8}  dr =\frac{24\pi G \rho}{3R^2}\left[\frac{r^4}{4} - \left(\frac{R^3}{3}\right)r\right]_{r=R/2}^{r=R} = \frac{11\pi G \rho}{8}R^2.$$
If we use $\rho=3.34$ g/cm$^3$ and $R=1737.1$ km I get $P= 2.9056$ MPa. That doesn't sound too bad for rock to bear. It is the weight a tunnel roof 110 meter underground would need to handle, and we certainly have those.
But there is a problem: all this force needs to be balanced by counteracting forces along the inner surface. One can view a hemisphere of the moon as a dome sitting on an imaginary plane. All downward force (in the direction of the plane) sums up and must be met by a corresponding force from the other side. The total force on a shell of thickness dr is $$F=P \int_0^{\pi/2}\int_0^{2\pi} (R/2)^2 \sin(\theta)\cos(\theta)d\psi d\theta dr = (\pi/8) R^2 P dr$$ and it is distributed over an area $A = \pi (R/2) dr$, giving a lateral pressure $F/A = R P/4$. Which in our case becomes 1.2618 TPa. That is way unrealistic for rock to hold up. 
We might quibble that lateral pressure further up helps reduce the overall pressure from the overlaying shells: after all, not everything is resting on the innermost shell. A more careful model of the elastic forces sounds like a decent homework exercise. The actual answer is likely a few orders of magnitude less. 
But the general conclusion is that there is going to be a lot of stress on the innermost shell. That means that if there is a single point of failure there will be a rock explosion where parts of the shell fly into the hollow interior. Once this happen neighbouring rock is unsupported, and so on until we have a collapsed moon. 
Since lunar rock has no quality control and already contains various cracks, the probability of a disaster along the interior surface is pretty guaranteed. So the hollow moon will not remain hollow for long. Still, sufficiently small bodies clearly can be made hollow: the moon is just a bit too large.
