Quartz sandstone (QS) is a mine waste; therefore, its use in construction allows for both reducing the cost of the concrete and contributing to the utilization of waste. The scientific originality of this study is the identification of models of the effect of QS aggregate on the physicomechanical, durability characteristics, and eco-safety of greener high-strength concrete. The study used an energy-efficient method of non-thermal effects of electromagnetic pulses on the destruction mechanisms of quartz-containing raw materials. The characteristics of quartzite sandstone aggregates, including the natural activity of radionuclides, were comprehensively studied. The features of concrete hardening, including the formation of an interfacial transition zone between the aggregate and the cement matrix, were studied, taking into account the chemical and morphological features of quartzite sandstone. In addition, the microstructural and morphological properties of concrete were determined after a 28 day curing. In this study, the behaviors of the concrete with QS aggregate were investigated, bearing in mind the provisions of geomimetics science on the affinity of structures. The results obtained showed that the QS aggregate had the activity of natural radionuclides 3–4 times lower compared to traditional aggregates. Efficient greener concrete with a 46.3 MPa compressive strength, water permeability grade W14, and freeze–thaw resistance of 300 cycles were also obtained, demonstrating that the performance of this greener concrete was comparable to that of traditional concrete with more expensive granite or gabbro diabase aggregates.