Yes, but your water has no buffer capacity. If I take demineralized water and I see a pH of 8 and I add a drop of acid, it can go to pH 2. Then if you add three drops of NaOH, it goes to pH 11. So what we do in the lab is mix a weak base or acid and it's salt that dissociates at a certain pH. If the concentration of the buffer is high, you can add many drops of sulfuric acid, and the pH won't move. The weak base and salt just move in equilibrium while the pH stays the same.
If your soil is very basic, it is so for a reason. There are minerals that act as weak bases at remove H+ from water, increasing the pH. Your water with a tiny bit of acid could have it's acid completely neutralized and your pH 6.0 water will have no effect at all on soil pH. This also doesn't mean that you should add even more acid and have a really low pH. Or try to buffer the water. I think it is more beneficial to remove CaCO3 ions from water, because they make it more basic and they would accumulate. You could still treat the soil with sulfur compounds, and that will generate sulfuric acid, and that will lower soil pH. Don't know the details about how to do that. But just adding a very concentrated acid could be very dangerous.
Now maybe I am overthinking this and the roots experience a nice pH 6 for as long as your water is there, because there is just that much more water than soil particles and basic minerals. That would explain why your azalea seem to do decent when the soil if it were actually pH 9 should be deadly. But my point still stands that once your water that is pH 6 hits the soil, it could suddenly be very different.
