This might be a way to look at it: Let's pose the question with the assumption that there is no gravity field, and let's also assume the "particles" (be they molecules or atoms) have spherical symmetry. Then no special direction is imposed on the system, so, the field of motion must be isotropic. If one were to insert an imaginary reference plane into the fluid, no matter what its orientation one must observe the same velocity spectrum for particles crossing this plane, thus, one "sees" the same momentum fluxes (in either direction across the reference plate) and thus the pressure field is isotropic and homogeneous.
So soon as we add gravity, a vertical gradient in pressure is required to balance the weight of any fluid element (as noted above by @David White). Fluid mechanics hinges on invoking the "continuum hypothesis", the idea that at each point of the medium there are well-defined values of pressure, density etc. Implicitly one has carried out a volume average over a length scale that is long compared to the mean free path of the constitutive particles (which at any instant are carrying momentum in every which direction), to obtain properties that vary continuously with the position coordinates. There is perhaps some loss of clarity when one regards fluid pressure as "pointing" in any given direction - though a pressure gradient certainly does.