This paper presents a novel system design for low earth orbit (LEO) satellite communication systems using stacked intelligent metasurface (SIM) technology. The lightweight and energy-efficient SIM is mounted on the satellite to achieve multiuser beamforming directly in the electromagnetic wave domain, significantly reducing processing delay and computational load compared to traditional digital beamforming. To address the challenge of obtaining instantaneous channel state information (CSI) at the transmitter, a joint power allocation and SIM phase shift optimization problem based on statistical CSI is formulated, 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 enhance system performance. Simulation results demonstrate the effectiveness of the proposed SIM-based LEO satellite system design and the statistical CSI-based AO algorithm, showing comparable performance to digital systems with reduced processing complexity and energy consumption.This paper presents a novel system design for low earth orbit (LEO) satellite communication systems using stacked intelligent metasurface (SIM) technology. The lightweight and energy-efficient SIM is mounted on the satellite to achieve multiuser beamforming directly in the electromagnetic wave domain, significantly reducing processing delay and computational load compared to traditional digital beamforming. To address the challenge of obtaining instantaneous channel state information (CSI) at the transmitter, a joint power allocation and SIM phase shift optimization problem based on statistical CSI is formulated, 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 enhance system performance. Simulation results demonstrate the effectiveness of the proposed SIM-based LEO satellite system design and the statistical CSI-based AO algorithm, showing comparable performance to digital systems with reduced processing complexity and energy consumption.