Unfortunately not, this was a famous problem taught by Feynman in his physics lectures. Essentially the Brownian motion of the ratchet and pawl mechanism (used in your watch) will eventually prevent the mechanism from harvesting energy [0].

[0] - https://en.m.wikipedia.org/wiki/Brownian_ratchet

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.

Macroscopic motions of the watch in random directions would probably still work.

Because even random macroscopic motion still has less entropy than a room temperature watch.

The problem is that once the particles in your watch bounce around at the same rate (it is in thermal equilibrium with the environment) then it's statistically impossible to induce a net flow of energy from the environment into the device.