This letter presents a novel system design for Low Earth Orbit (LEO) satellite communication systems using stacked intelligent metasurfaces (SIM). The SIM technology enables lightweight and energy-efficient multiuser beamforming directly in the electromagnetic wave domain, significantly reducing processing delay and computational load compared to traditional digital beamforming. To address the challenges of obtaining instantaneous channel state information (CSI), a joint power allocation and SIM phase shift optimization problem is formulated based on statistical CSI, and an alternating optimization (AO) algorithm is customized to solve it efficiently. Additionally, a user grouping method based on channel correlation and an antenna selection algorithm are proposed to further improve system performance. Simulation results demonstrate the effectiveness of the proposed SIM-based LEO satellite system design and statistical CSI-based AO algorithm. The system achieves performance comparable to digital systems without the need for digital precoding, and using statistical CSI approaches the performance of instantaneous CSI. The gains are attributed to the devised grouping and antenna selection methods. The proposed system uses statistical CSI to maximize the ergodic sum rate by optimizing SIM phase shifts and power allocation. The system also employs user grouping based on channel correlation and antenna selection to minimize total leakage energy, enhancing performance for large numbers of users. The simulation results show that the proposed SIM-based system outperforms benchmarks, achieving higher sum rates with reduced computational complexity. The system is designed to operate in the wave domain, providing lower processing latency and computational burden compared to traditional digital precoding methods. The results demonstrate the effectiveness of the proposed SIM-based design and the customized AO algorithm in enhancing LEO satellite communication performance.This letter presents a novel system design for Low Earth Orbit (LEO) satellite communication systems using stacked intelligent metasurfaces (SIM). The SIM technology enables lightweight and energy-efficient multiuser beamforming directly in the electromagnetic wave domain, significantly reducing processing delay and computational load compared to traditional digital beamforming. To address the challenges of obtaining instantaneous channel state information (CSI), a joint power allocation and SIM phase shift optimization problem is formulated based on statistical CSI, and an alternating optimization (AO) algorithm is customized to solve it efficiently. Additionally, a user grouping method based on channel correlation and an antenna selection algorithm are proposed to further improve system performance. Simulation results demonstrate the effectiveness of the proposed SIM-based LEO satellite system design and statistical CSI-based AO algorithm. The system achieves performance comparable to digital systems without the need for digital precoding, and using statistical CSI approaches the performance of instantaneous CSI. The gains are attributed to the devised grouping and antenna selection methods. The proposed system uses statistical CSI to maximize the ergodic sum rate by optimizing SIM phase shifts and power allocation. The system also employs user grouping based on channel correlation and antenna selection to minimize total leakage energy, enhancing performance for large numbers of users. The simulation results show that the proposed SIM-based system outperforms benchmarks, achieving higher sum rates with reduced computational complexity. The system is designed to operate in the wave domain, providing lower processing latency and computational burden compared to traditional digital precoding methods. The results demonstrate the effectiveness of the proposed SIM-based design and the customized AO algorithm in enhancing LEO satellite communication performance.