Not quite true. It IS true that most distilled water has an acidic pH (that is, below 7.0). That is because it is so very pure, but has a small amount of normal air dissolved in it. Now the two major components of air, nitrogen and oxygen, do nothing unusual in distilled water. But the third one, carbon dioxide (normally about 4% of air) reacts with the water to produce carbonic acid, a weak acid which partially dissociates, releasing free hydrogen ions and tying up the associated OH groups with the CO2. It is those free H+ ions that are called acid, and show up in a pH test. However, the amount of this is VERY low - it only shows as reduced pH because there is almost nothing else in the very pure distilled water to balance it out. The amount of dissolved CO2 in the water is absolutely harmless to people. If you are really worried about it, just boil the distilled water for a couple of minutes, then pour it into a container with no air and cool it down. Boiling the water will force all the dissolved air out, and then you just seal it up with no access to air. But I would never bother.
The more interesting issue is, why does anyone think the absolutely (chemically) pure water, or distilled water, is healthy? It certainly is NOT "natural". People have evolved with a variety of water sources to drink, and NONE of them have been close to distilled in "purity". While I grant you that distilled (a product of modern technology) is better than some waters that are contaminated with new man-made pollutants, it is NOT a substitue for clean normal drinking water. I agree with wyvrn that the mineral content of normal water is more important.