Depending upon where one gets his drinking water from, most tap water is lightly basic pH around 8 IIRC to prevent corrosion of home plumbing. IMO it would be mostly useless for removing bear spray. Something mildly acidic might assist in rendering it more water soluble. Also if there is some kind of oil in the stuff then that doesn't help.
A place I used to work at used a basic aromatic amine type compound that is a skin irritant. A number of people got sensitized to it. Generally it was used in a salt for. If they had any on them and took a shower the basic water ended up releasing enough of the free base to cause some people problems.
I assume the guy eventually recovered with no permanent damage.
You've piqued my curiosity. I presently reside on Washington's Olympic Peninsula (I hope to be hunting Montana on a resident license next year as I recently purchased a nice rural property east of Missoula). Here, the ground long covered by mostly coniferous forests generally tends to the acidic side of the pH scale. I would be surprised to find that my tap water tends toward the base. Now, I must shop pH test strip availability and prices online to see if I can justify an experiment!
I also assume that the gentleman fully recovered without lingering deleterious impacts. Other than still evident skin reddening and bloodshot eyes, he seemed substantially normal by the time we concluded the approximately 8-mile hike out to the trailhead.