Interference management is a central bottleneck in dense multi-antenna wireless networks. Therefore, in this study, we present a digital precoding-free hierarchical rate-splitting multiple access (HRSMA) architecture assisted by a stacked intelligent metasurface (SIM) to achieve high spectral efficiency and user fairness with reduced hardware complexity. In the proposed system, the base station performs only scalar power allocation, while a multi-layer SIM acts as a wave-domain processor that spatially separates users and mitigates interference via nonlinear wavefront reconfiguration. This design eliminates the need for digital or hybrid precoding, drastically reducing the baseband computations. A joint optimization problem is formulated to maximize the minimum user rate by jointly optimizing SIM phase shifts, power allocation, and user grouping. To efficiently solve the resulting non-convex problem, an alternating optimization algorithm is developed, combining simultaneous perturbation stochastic approximation (SPSA) for SIM configuration and power control with clustering-based grouping refinement. Simulation results demonstrate that the proposed SIM-aided HRSMA achieves substantial gains in both spectral efficiency and fairness compared to hybrid beamforming and non-precoding baselines. Specifically, SIM-aided HRSMA attains comparable or superior minimum rates with significantly fewer active antennas by exploiting the additional wave-domain degrees of freedom provided by multi-layer SIMs. These findings highlight the potential of SIM-aided HRSMA as a low-cost, energy-efficient, and scalable solution for beyond-6G networks.