The motion that winds a self-winding mechanical clock is the swinging motion of an arm. Swinging motion is directional, back and forth, i.e. predictable. Indeed, the very self-winding mechanism is possible only because of this predictable movement.

Brownian motion is literally random diffusion. By comparison, it'd be like vibrating the watch in every direction. It wouldn't wind from such motion. No known mechanism exists that would.