Of course condensation will cause water spots and hard water spots as well, maybe not to the same extent but absolutely it will. How do you have a shower without a vent? That, in the long run will cause you plenty of other issues. If I remember correctly this is a new house too, right? Does the glass run all the way to the ceiling?

Unless the relative humidity in the air is from distilled water (it isn't), there will be mineral deposits.

" I know that minerals in the water cause water spots so it seems logical that condensation won't have minerals "

Is that some sort of new kind of logic? Condensation is water, and your water has minerals in it. So normal logic dictates that the condensation will have minerals.

Theoretically, condensation will be pure water with no minerals present. This is high school chemistry stuff, minerals in water don't boil off into the produced steam that then forms condensation. Can vigorously produced condensation (steam) carry minerals, yes, a vigorously boiling steam generator can entrain some minerals in the output stream. And, water splashes in showers do happen! I would play it safe and squeegee everything, if only to help the bath to dry faster to protect the structure from long term water damage and mold issues.