Why are there clear-cut states of matter instead of a gradual transition from gas to solid (let's set plasma aside for the purpose of this question)? If the main difference between them is the distance between molecules, then with temperature going down (let's neglect pressure for the purpose of this question), a gaseous substance should gradually become more and more liquid and then more and more solid, shouldn't it (as if God fiddles with a slider in his computer interface; let's set aside the question of God's existence for the purpose of this question)?
Some substances do have a state that is intermediate between a solid and a liquid over a range of temperatures and pressures - its is called a mesophase. Liquid crystal displays are an application of mesophase materials.
This is a long comment:
A simpler question is: Can one have a mathematical theory of solid state matter without quantum mechanics?
The short answer is: No, lattices in solids can only be be modeled with quantum mechanics. This means there is no continuum of states between, example, water and ice.
Ice forms at a fixed temperature where the H2O molecules can quantum mechanically attract each other into a lattice.
For the other states you ask, the particular case can be studied , for example gas to liquid, and I expect the answer, though not so simple, will be the same, allowing for the intermediate case as discussed in the answer by gandalf61.