With Mars' color coming from the iron oxide, is it possible for such a small body without an atmosphere to support that color?
The redness of Mars' surface comes from the large quantities of iron in its crust. Other terrestrial planets in the Solar System also contain iron, but not at their surfaces. Mars in particular has several things going for it that allowed it to keep much of that iron in the crust:
These both prevented differentiation - essentially, the vertical reshuffling of different minerals due to differences in weight - and the melting of iron. Both would have caused the iron to drop into the planet's mantle and core. However, the environment wasn't conducive to this restructuring, and so the iron was able to stay in the crust. Your moon - presumably with a low mass - would have the same optimal physical properties, although, as I'll talk about later, the lack of an atmosphere is problematic.
Chemically, Mars is red because iron is present in the dust layer coating much of the surface. A hematite (Fe2O3) is likely the main culprit for the color, with much of the rest of the iron being stored in a black magnetite (Fe3O4). Now, one way to generate this involves oxygen gas (O2). However, oxygen was never present on Mars in large quantities - and it clearly can't be on a body with no atmosphere!
It's been suggested that a reaction of iron (Fe2+ or Fe) with liquid water could indirectly form the proper minerals. Dissolved carbon dioxide and sulfur dioxide (CO2 and SO2) could act to speed the reactions up. Hydrogen peroxide (H2O2) leads to yet another reaction pathway.
However, these two avenues both involve either an atmosphere or liquid water, which bother me for your specific setup. Now, it's been theorized that specific erosional processes could turn magnetite into the right kind of hematite. The crushing and colliding of magnetite with other materials provides a possibility, and doesn't necessarily require erosion by fluids (gases or liquids). It does, however, require a source for the magnetite in the first place.
All of that said, it's unlikely for hematite to easily form on a body with no atmosphere (and thus likely no liquid water). It's possible for it to form slowly, and on long geologic timescales, so I won't rule it out entirely. It would just be difficult, with a normal concentration of iron. It's possible that an iron-rich impactor - such as another protoplanet - could have collided with your moon early in its history, dumping iron all over the surface. The problem here is that collisions between celestial bodies tend to turn crusts molten - and as I said before, that's not good for keeping iron in the crust! It would cause extreme restructuring, and that's a problem. Could it happen? Maybe. Am I optimistic? Not terribly.
Inspiration from our Solar System
Let's look to our own planetary system for some inspiration. Mercury has a large amount of iron in its core - and in fact, Mercury's core is more than half of the planet, by volume. Its crust and mantle are comparatively thin. One common explanation for this is that an impactor stripped away most of the outer layers, leaving behind a planet that was, by mass, a greater percentage iron. Now, the iron in the planet was already in the core, and I would expect that the molten surface caused by such an impact would have allowed any additional iron to travel to the core.
Given that your moon is so small, I suspect that a similar impact, rather than a relatively grazing one, like the one Earth suffered, would have led to the same sort of process. Surface conditions would not have been good for iron to stay there, and it's certain that no iron would have been dredged up from the depths of the planet.
Now, a possible loophole is to have an event - or rather a series of events - like the Late Heavy Bombardment, which may have occurred about 4 billion years ago in the Solar System. A substantial number of asteroids and other bodies collided with the terrestrial planets (a similar event may be occurring in the exoplanetary system around Eta Corvi. A prominent theory for the cause of the LHB is the change in orbits of the giant planets in the outer section of the Solar System, as per the Nice model and variants thereof (e.g. the Nice II model). If an unusually large number of iron-rich meteorites struck your moon during an analogous event, you could have large-scale iron deposition without a single cataclysmic impact.
You want a red moon, and you note Mars is red, but you did not mandate that the red moon must be red for the same reason as Mars. Paint it red with tholins! http://www.planetary.org/blogs/guest-blogs/2015/0722-what-in-the-worlds-are-tholins.html
Consider Titan. It is a reddish orange moon. https://www.nasa.gov/mission_pages/cassini/multimedia/pia06081.html
Titan is a natty orange due to tholins in its atmosphere. http://www.planetary.org/blogs/guest-blogs/2015/0722-what-in-the-worlds-are-tholins.html
Europa has big red scratches on it that are probably red for the same reason (full of red tholins). And it turns out large swaths of Pluto (which is smaller than Titan or Luna) are also red from tholins.
The surface also has many dark, reddish patches due to the presence of tholins, which are created by charged particles from the Sun interacting with mixtures of methane and nitrogen
Or Mercury-Sulfur (HgS). The creation of cinnabar requires no atmosphere, just igneous rocks with some water. If the moon had a lively volcanic past and plenty of mercury and sulfur then it would be easy to get large patches of the lovely red powder you see below.
The question implies but does not require iron oxide.
So your question is basically, can an atmospheres moon have oxidized iron?
Short answer is yes.
Long answer depends on what the moon is made out of or how it formed. During the formation of our solar system there was a lot of gas floating around in the protostellar nebula. Could oxidation happen then? Why not.
At temperatures over 200°C, sulphurs turns a blood red colour.
Sulphur is the tenth most abundant element in the universe (by mass), and the fifth on Earth. Having a moon whose surface (or even entirety) is sulphur is not far fetched (Jupiter’s moon Io is covered with sulphur compounds).
Now the heat. Luna has a daytime surface temperature of about 100°C. To get a further 100°C, you could add:
- heating due to radioactivity at the moon’s core
- a brighter/hotter sun
- friction from tidal forces due to low orbit and/or other moons
Is it possible? Yes.
Here is a scenario. An asteroid collides with mars and shaves off enough of the surface layer (red) to make an object roughly the size of the moon. Now put rockets on it and move it into orbit around earth, or have it naturally come into orbit around something like the planet Earth.
Now you have a red moon. Over time it should even become sphere like.
You only asked if it was possible.