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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?

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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

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I'm not so sure that Solar wind is always the dominant factor. There is also the possibility that thermal fluctuations in the molecules' speeds can take them above the escape velocity. This would be the case on Earth if our atmosphere was made of $\text{H}_2$, IIRC. – Nathaniel Jan 30 at 14:05
good point, @Nathaniel – lurscher Jan 30 at 19:22
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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:-)

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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;-)

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The speed you give is (I think) the RMS speed, but there's a distribution of speeds around it. Some molecules will have higher speeds, and may escape, and if the temperature is maintained some other molecules will gain energy to fill those high velocity states in the distribution... leading to a gradual evaporation of the atmosphere. – Kyle Jan 30 at 14:30

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