The stomach acid (does) react with the water we drink.
The stomach pH value is maintained at around 4. When we eat food and drink water, especially alkaline water, the pH value inside the stomach goes up. When this happens, there is a feedback mechanism in our stomach to detect this and commands the stomach wall to secrete more hydrochloric acid into the stomach to bring the pH value back to 4. So the stomach becomes acidic again. When we drink more alkaline water, more hydrochloric acid is secreted to maintain the stomach pH value. It seems like a losing battle. However, when you understand how the stomach wall makes hydrochloric acid, your concerns will disappear.
However, there is no hydrochloric acid pouch in our body. If there were, it would burn a hole in our body. The cells in our stomach wall must produce it on an instantly-as-needed basis. The ingredients in the stomach cell that make hydrochloric acid (HCl) are carbon dioxide (CO2), water (H2O), and sodium chloride (NaCl) or potassium chloride (KCl).
NaCl + H2O + CO2 = HCl + NaHCO3, or
KCl + H2O + CO2 = HCl + KHCO3
As we can see, the byproduct of making hydrochloric acid is sodium bicarbonate (NaHCO3) or potassium bicarbonate (KHCO3), which goes into blood stream. These bicarbonates are the alkaline buffers that neutralize excess acids in the blood; they dissolve solid acid wastes into liquid form. As they neutralize the solid acidic wastes, extra carbon dioxide is released, which is discharged through the lungs. As our body gets old, these alkaline buffers get low; this phenomenon is called acidosis. This is a natural occurrence as our body accumulates more acidic waste products. There is, therefore, a relationship between the aging process and the accumulation of acids.
By looking at the pH value of the stomach alone, it seems that alkaline water never reaches the body. But when you look at the whole body, there is a net gain of alkalinity as we drink alkaline water. So, water does have some reaction with the stomach acid or the other way around.