It all comes down to how long the spots sat on the paint, what was in the water, and how soft/hard the paint is.
I've had personal experience on both ends of the spectrum: a quick wash with a 50/50 water/distilled white vinegar wash took 'em right off one car but on another even three passes with a high speed rotary and a fairly aggressive compound on a wool pad couldn't get rid of all of them.
A clay bar won't do anything if the paint is actually etched as clay will only remove bonded surface contaminants. Stubborn deposits can be removed with clay, but true etching is below the surface of the clear - that is, some of the clear is literally gone and the only way to "remove" the spot is to actually remove some of the surrounding clear. Sounds bad, I know, but facts is facts. And the fact is, sometimes the damage is bad enough that without really drastic measures (wetsanding and compounding) you can only minimize the appearance.