Do celestial objects need to be big to have liquid water on their surfaces?
In a nutshell: liquid surface water needs an atmosphere. To sustain an atmosphere, a planet must be sufficiently massive, therefore sufficiently large. The warmer a planet, the more mass it needs to sustain an atmosphere. A planet warm enough for liquid water must thus also be large enough to sustain the atmosphere for this liquid surface water to survive.
Liquid water can only exist if pressure is larger than 612 Pa. The boiling point depends on the pressure. At a pressure of 101 kPa, like average conditions on Earth sea level, the boiling point is 373 K (100 °C). At a pressure of 34 kPa (average on the summit of Sagarmatha/Mount Everest), the boiling point is 71 °C:
Source: Cmglee, Wikimedia Commons
For liquid water to exist at the surface, a sufficiently thick atmosphere must exist to provide this pressure. It could be a little thinner than on Earth, but if it's too thin then water might boil all too easily.
Mars has an average surface pressure of 636 Pa on average, which means that in theory, liquid water could barely exist, but only when the temperature is pretty much exactly 273 K (0 °C). One degree colder and it will freeze, one degree warmer and it will boil. In reality surface temperature on Mars is on average 210 K. The pressure on Mars depends on the location, but for future Mars colonists, it should be fun challenge to try to see how long they can make liquid water survive (heated, but unpressurised) at some of its lowest points!
To retain an atmosphere, a planet must have sufficient gravity. If there is not enough gravity, most of the atmosphere will drift off into space due to thermal effects (molecular thermal velocity in excess of escape velocity, see below), solar wind escape (charged particles pushing against the atmosphere), and other (smaller) effects. Worse, not only is most of the atmosphere lost if a planet is too small, but the lighter species such as water are lost more easily than, say, carbon dioxide. So not only must gravity be sufficient to hold onto an atmosphere, it must be sufficient to hold onto water specifically. The only way to counter or prevent losses to space are sufficient gravity (reducing loss) or a constant new supply. The size a planet needs to have to retain an atmosphere depends on temperature:
Source: Cmglee, Wikimedia Commons
To have sufficient gravity, a planet must have sufficient mass. To have sufficient mass, it must be sufficiently large. How large is "sufficiently large"? That depends on the temperature. Titan is quite small, but also very cold. At temperatures warm enough to sustain water, a planet could be a little smaller than Earth, but not much. Mars is too small. Although a Mars-sized planet at an Earth-like temperature could theoretically retain a Titan-like atmosphere of mostly nitrogen for a while (it'd be close, being a bit larger would be safer), it would still lose its water over time.
To have a constant new supply, a planet or moon would need vulcanism. To sustain vulcanism, a planet needs an internal heat supply, for which it also needs sufficient mass, at least to sustain this long term. A moon can also get energy from a planet to sustain volcanism. Maybe a warmer hybrid between heavily volcanic Io and Enceladus with cryovolcanoes around a hypothetical extrasolar planet could sustain a highly dynamic atmosphere, even if it would normally be too small according to the diagram above. That might be unlikely, though; in case of Io, the same energy source that powers the volcanism also strips away the atmosphere (and Io has the least water of anywhere in the solar system). In any case, the only moon with a significant atmosphere in our solar system is Titan, which is also the smallest body in the solar system with an atmosphere. It's very cold at 94 K; if it were warm enough to contain liquid water, it would lose its atmosphere.
Maybe a very young planet could be quite small, still vulcanically active, and still keep on to enough atmosphere to allow for widespread liquid surface water. There are no such planets in the solar system either, but it could be conceivable for an extrasolar planet. For any planet of significant age, however, only mass, thus only size, will help.
Size does matter.