By proving a weighted contraction estimate in uniformly local Sobolev spaces for the flow of gravity water waves, we show that this nonlocal system is in fact pseudo-local in the following sense: locally in time, the dynamic far away from a given bounded region has a small effect on that region (again, in a sense that we will make precise in the article). Our estimate on the flow also implies a new spatial decay property of the waves. To prove this result, we establish a paradifferential calculus theory in uniformly local Sobolev spaces with weights.