This is not true, not in any meaningful sense.
First of all, non-distilled water has extremely widely varying levels of various minerals; for example among U.S. cities magnesium concentration varies by 300-fold (and other minerals also vary widely). Since leeching is proportional to concentration difference, the difference between zero and a tiny trace is much less chemically significant than the difference between a tiny trace and a 300-fold larger trace.
Second of all, the minerals in drinking water are, for a balanced diet, overwhelmed by the minerals present in food. For example, in the U.S. cities list, if you drink 2L of water of the highest mineral concentration, you would get 120, 180, 220, and 30 mg of magnesium, calcium, sodium, and potassium, respectively. The daily recommended intake (linked from here) for 19-30 year old males is 400, 1000, 1500, and 4700 mg. These differences are vast, and are similarly large for the other minerals I've checked. Certainly, since water has no memory, drinking distilled water wouldn't take away more than 120 mg of magnesium beyond what drinking the highest magnesium concentration tap water in the U.S., and this is a negligible amount compared to the daily intake.
However, if one has poor nutrition to begin with, water may provide (no guarantees, given the large variations!) enough of some mineral to alleviate negative impacts on health. Still, that distilled water is "removing minerals" is not a useful way to think about it. Yes, of course when we excrete waste products, some minerals go along; if you're short of some mineral, you'll then be even shorter. If you're lucky enough to have the deficiency covered by the water you drink, then you'll of course do better if you don't change water supplies. But the mineralization of water is widely variable, and it is not the distinction between distilled and undistilled but the particular concentration of particular minerals in particular water that is important. For example, if you're desperately short of calcium, 180 mg/day from water is going to be a lot better for you than 0 mg/day!
Short version: mineral content in water has a minimal to negligible impact in the context of good nutrition; it is possible for there to be enough of a mineral in a water supply to make a difference in the context of poor nutrition.