If astronauts could deliver a large quantity of breathable air to somewhere with lower gravity, such as Earth's moon, would the air form an atmosphere, or would it float away and disappear? Is there a minimum amount of gravity necessary to trap a breathable atmosphere on a planet?
Gravity is a major factor in planets retaining atmospheres over the eons. But there are other factors that must be taken into consideration to consider the volatility of an atmosphere.
Solar wind is the main factor of erosion on any atmosphere. But a healthy magnetic field can deflect most of the solar radiation and decrease the erosion. It has been a matter of debate recently if exo-moons of jovian planets in habitable zones of their host stars would be able to sustain atmospheres: such moons are most likely tidally-locked, so their magnetic fields are not expected to be high, but their host planets will likely have strong radiation belts. But is not clear at the moment if the radiation belts will protect or erode the atmosphere. Saturn has a benign level of radiation, so we have Titan, which has an atmosphere that is thicker than earth's
The escape velocity at the moon's surface is about 2.4 km/s. The mean speed of oxygen at 293 K is about 0.48 km/s.
A commonly quoted rule of thumb says that the escape velocity needs to be 6 times the gas's mean velocity in order for that gas to remain captive to gravity and the values I quoted are related by a factor of only 5. The air would contain water (since dry air is very uncomfortable to breath) and carbon dioxide (as a by-product if not also needed to sustain the cyanobacteria/plants you would want in place of planetary size mechanical carbon dioxide scrubbers, then there are the nutrients you would need to sustain those) which would readily exacerbate an atmospheric greenhouse effect and, with the moon being at about the same distance from the sun as is earth, you would expect the air to warm up to similar to earth temperatures, though without the moderating effect of oceans, and so cause the oxygen to dissipate. As nitrogen is lighter it's mean speed at the same temp is higher, v_rms something like 0.51 km/s IIRC, so it too would dissipate as would water vapour.
In short, it doesn't seem likely that it would be possible on the moon.
As an aside, ignoring for a moment shielding from the solar wind noted by turscher, Venus and Earth have similar surface gravities but Venus' atmosphere is much thicker than Earth's so gravity is not the sole factor in determining atmospheric retention and neither is temperature as Venus is very much hotter than Earth.
To answer the part of your question about a minimum gravity needed which no one else seems to have addressed: Surface gravity would have to be such that it requires a escape velocity around, as that rule of thumb states, six times the v_rms of any gases you wished to retain. With a too low escape velocity over time gases will escape, lighter gases first, leading to a thinning of the atmosphere and a time-dependent composition. But this could take geologic ages. Any particular loss process could be so slow that it would be easily replenished by whatever process the astronauts used to create the atmosphere. If part of that process was bombardment by comets (largely for their water content) care would have to be taken as such extra-planetary bombardment could also be very damaging to a planet's atmosphere.
I guess the devil is in the details. For example, if the celestial body in question is far from its star, so its temperature is very low, it is easier to retain low-temperature air around the body. On the other hand, very cold air is not breathable anyway. There is another way though. If the astronauts can bring so much air to the body, why don't they arrange a membrane around the body to keep the air? Furthermore, they don't need the membrane around the entire body, they can arrange it just over some limited area where they want to live. On the other hand, they would need to protect such a membrane from meteorites... So I guess there is a lot they can do and a lot of factors that could make their life miserable:-)
The moon has 85% of the gravity of Titan (which has a thick hydrocarbon atmosphere), so I cannot believe for 1 second that it's gravity is too weak to retain a viable atmosphere.
Factors like Sola winds stripping the atmosphere due to lack of protection from a magnetic field, is a valid explanation, but low gravity cannot be, because the existence of Titan disproves that.
The speed of oxygen at room temperature (293k) is 1720km per hour so if the escape velocity of the moon or planet is greater than that then at least you will have oxygen. If you want some nitrogen in the mix then you will have to google it's speed like I did for oxygen;-)