Computed tomography (CT) reconstruction with sparse-view projections is a challenging problem in medical imaging. The learning-based methods lack generalization ability and mathematical interpretability. Since the model-based iterative reconstruction (IR) methods need inner gradient-based iterations to deal with the CT system matrix, the algorithms may not be efficient enough, and IR methods with deep networks have no convergence guarantees. In this paper, we propose an efficient deep semi-proximal iterative method (DeepSPIM) to reconstruct CT images from sparse-view projections. Unlike the existing IR methods, a carefully designed semi-proximal term is introduced to make the system matrix-related subproblem solvable. Theoretically, we give some useful mathematical analysis, including the existence of the solutions to the reconstruction model with an implicit image prior, the global convergence of the proposed method under gradient step denoiser assumption. Experimental results show that DeepSPIM is efficient and outperforms the closely related state-of-the-art methods regarding quantitative image quality values, details preservation, and structure recovery.