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This paper proposes a graph neural network (GNN)-based space multiple-input multiple-output (MIMO) framework, named GSM, for direct-to-cell communications, aiming to achieve distributed coordinated beamforming for low Earth orbit (LEO) satellites. Firstly, a system model for LEO multi-satellite communications is established, where multiple LEO satellites collaborate to perform distributed beamforming and communicate with terrestrial user terminals coherently. Based on the system model, a weighted sum rate maximization problem is formulated. Secondly, a GNN-based method is developed to address the optimization problem. Particularly, the adopted neural network is composed of multiple identical GNNs, which are trained together and then deployed individually on each LEO satellite. Finally, the trained GNN is quantized and deployed on a field-programmable gate array (FPGA) to accelerate the inference by customizing the microarchitecture. Simulation results demonstrate that the proposed GNN scheme outperforms the benchmark ones including maximum ratio transmission, zero forcing and minimum mean square error. Furthermore, experimental results show that the FPGA-based accelerator achieves remarkably low inference latency, ranging from 3.863 to 5.883 ms under a 10-ns target clock period with 8-bit fixed-point data representation.