Due to complex yarn geometries and multiscale nature, it is very challenging to predict mechanical properties and strength constants of woven composites in an accuracy and efficient way. In this paper, mechanics of structure genome (MSG) is employed to predict effective material properties and strength constants of woven composites. The microscale analysis is performed to compute the effective properties and strength constants of yarns based on a hexagonal packed microstructure. Then, the mesoscale analysis is performed to compute the effective material properties and strength constants of woven composites. The accuracy and efficiency of the MSG method are compared with the computational homogenization method. In addition, simulation data using the MSG model is generated to train deep learning neural networks (DNN) models. The trained models provide ultra-efficient surrogate models with high accuracy. The DNN models are further applied to a design optimization problem to tailor geometry parameters to enhance the mechanical performance of plain woven composites, and the optimization can be efficiently solved with gradients from the DNN models. The developed computational framework can be applied to general woven composites featuring complex microscale and mesoscale features.