Q: Are rocks and other solid objects just very slow moving fluids? Will a rock or other solid object under the influence of gravity, given enough time, behave like a fluid and slowly flatten out? Or do the properties of matter that make it solid prevent it from ever deforming in the way a fluid does?

A: There is no such thing as a perfectly rigid body. The rigidness of a body is given from the chemical bonds that hold the object together, so given enough energy you can deform any material without liquefying it.
There is no real boundary between a really viscous fluid and a really soft solid, as the famous experiment of the pitch drops shows.
This was of course very general, while your answer is quite specific. There exist countless cases of deformed rocks, usually due to mountains formation, as in this image, but the pressure that curves this rocks is much larger than the one exerted by gravity alone. On the surface of Earth, gravity can't slowly deform rocks, it doesn't have enough energy to rearrange the chemical bonds.

A: In continuum mechanics, fluid is defined (for e.g. see Fluid dynamics by Batchelor) to be that form of matter which $\textit{deforms continuously}$ so far as a $\textit{shear stress}$ acts on it, $\textit{no matter how small}$ the applied shear stress is. As per this definition, rocks do not qualify as fluids. You may object that it is an arbitrary definition and so it is, but you need some such definition, because you can make anything deform continuously with an appropriate amount of stress.

A: Crystals (rocks, metals, etc.) do not flow (unless they are heated to melting). Glass is amorphous (not crystal), but it doesn't flow either (in spite of the "urban legend" that says very old glass has "drooped".
Ice in glaciers is said to "flow", but it's not clear if that is any different from simple plastic deformation (i.e. movement of crystal dislocations (video)), like you get when you bend metal.
I can't say there are no solids that don't flow.
Butter, maybe?

A: 
Will a rock or other solid object under the influence of gravity, given enough time, behave like a fluid and slowly flatten out? Or do the properties of matter that make it solid prevent it from ever deforming in the way a fluid does?

Samples of Moon rock taken from the surface  of the moon by the Apollo crews are estimated to be around 4.5 billion years old, so they are a few hundred million years older than the oldest Earth rocks.
Assuming that geological processes are slower and less violent on the moon, and that possibly less repeated heating and cooling occurred,  there is a greater probability that these lumps or rock have not been affected by   gravitional slumping for a long,  long time.
If gravity has not caused slumping for that length of time, it could be safe to extrapolate to say it will never occur.
