Why doesn't the solute settle out of a solution, such as brine or sugar water?
You discuss two different phenomena, solubility and concentration gradients due to gravity. 

Solubility can be explained using the concept of entropy. Solubility requires the the free energy of a solid be higher than the corresponding substance dispersed in solution. Surprinsingly perhaps, additional entropy associated with dispersion of the solute in the solvent suffices to drive the process (interactions can certainly be important, but are not at minimum necessary, although much depends on how much you are attempting to solubilize, and at what temperature).

Gravity can (and does) cause concentration gradients, and can therefore under the right circumstances cause precipitation. The problem is that gravity is a very weak force, so it's effects on light molecules are, well, weak. Gravitational effects require large scales or masses to be significant. Density differences are argued to be critical driving force behind water cycling in the oceans. The air atop Mt Everest is so thin people suffocate without access to an auxiliary supply of oxygen. So these effects do happen, and matter. And you can use a centrifuge to separate components in a mixture, which is analogous to applying gravitational forces, just scaled up in strength.