Since life on Earth is carbon based, minerals that are bound with carbon are generally considered organic, while those bound with other elements are inorganic. This is oversimplified, but is a good rule of thumb.
The juices in fruits do, in a sense, contain distilled water. It is better to say the juice contains pure water and minerals that have been restructured by the plant to a smaller molecules in an organic form. The minerals you get from plant sources are those that the body can utilize on a cellular level. Most of the dissolved rock minerals are too large to enter into the tiny capillaries that feed the cells. Those that do make it into the blood stream are often deposited in joints or wind up lining the blood vessels making them narrow and brittle.
People who support the use of distilled water are not saying that minerals are not important, but they realize that not all mineral molecules are created equl, and not all can be assimilated by the body.