I have just completed a lot of research on this subject. The answer is no. Distilled water cannot remove nutrients from your body. I have 24 doctors reports confirming how healthy distilled water really is. //www.curezone.org/forums/fm.asp?i=2000592
We have well water that we thought was healthy. Unfortunately ground water accumulates toxins from farming, acid rain, defoliating. You should see the CRAP that is left behind in the distiller. Smell it to see the foul odor. There is much more than minerals left behind in our distiller.
After 1 month, we LOVE our distilled water, we use it in coffee, tea and drinking water. The wife & I drink about 1 gallon a day.
To answer your question, we need 1/2 teaspoon of natural salt and it would be great to add it your your distilled water. I like to put the 1/2 tsp of salt in my coffee.