It certainly can, and in my opinion, it happens to a very limited extent. If we want anything to leave the Earth’s gravitational field, we have to give it the so-called 2nd escape velocity (or Ve). Everything we throw/fly slower will come back or will orbit the Earth in an ellipse. The value of this speed is (at the Earth’s surface) 11.2 km/s. This is quite a high speed, of course the higher we are, the higher the value of 2 Ve will decrease but we will still be moving in high values. For example, at an altitude of 100 km, which is a kind of limit of space, the value 2 Ve is approximately 10 km/h.

It is therefore obvious that if the water molecule has a speed greater than 2 Ve, it can leave Earth’s gravitational field and move through space. But is it possible for a water molecule to ever reach such a speed?

This (as well as other physical phenomena) was dealt with by J. C. Maxwell and, using statistical methods, he derived his (Maxwell’s) distribution of molecules according to velocities. In this approach, Maxwell did not treat substances (e.g. water vapour in the atmosphere) as individual molecules but determined the probabilities of certain velocities. Compared to classical physics, it is a completely different approach (and it was also not very appealing to the physicists of the time — see the death of Ludwig Boltzmann). Instead of individual particles, we use statistics to solve a set of a huge number of particles. This approach really works and thanks to it we can calculate that at a temperature of 30 °C, the most likely speed of air molecules is about 560 m/s. This is certainly not enough to get the molecules out of the Earth’s gravitational field. But thanks to the randomness of the movement of particles, when their speed changes rapidly due to mutual collisions, there is a (very) small probability that the molecule will acquire a speed greater than 2 Ve and flies off into space. This probability (if I count correctly) is 0.0000000001%. The lighter the molecule, the easier it is to reach the required speed and therefore we lose hydrogen and helium rather than water.

Incidentally, Maxwell’s distribution of velocities also explains why the Moon has no atmosphere.