Depends on what you mean by below.
For temperature below the temperature of triple point ##T_c##, there can be equilibrium of solid with liquid, provided the pressure is high enough, although the region reaches only few degrees below ##T_c##.
See the phase diagram here:
http://www.lsbu.ac.uk/water/phase.html
For the pressure below the pressure of triple point ##p_c##, indeed there can be no liquid. Instead, there can be only solid, gas, or both.
Why gas and not liquid, is a good question. I do not know the full answer from molecular point of view, but here is what I know.
In principle liquids always exist thanks to some external pressure. In lake there is liquid water thanks to the atmospheric pressure - without atmosphere, the water would start to boil and evaporate away.
Liquids are always prepared to evaporate, since their molecules move chaotically and there are always some that have necessary escape velocity. So there has to be something that prevents that escape, a vessel of finite volume or the atmospheric pressure great enough to prevent boiling.
It turns out that for liquid water, the lowest pressure capable to do that is 612 Pa; the character of forces between the water molecules is such that lower pressure is simply not sufficient to keep the molecules tied together in liquid form. They either evaporate, or become part of the crystal, where they are kept by appreciably stronger forces than in the liquid.
(If the pressure is lowered down further, even the solid crystal can turn into gas, but this has limit, since below 200 K there seems to be no gas possible. (Again, hard to explain why exactly...))
