A number of recent studies have focused on developing surgical simulation platforms to train machine learning (ML) agents or models with synthetic data for surgical assistance. While existing platforms excel at tasks such as rigid body manipulation and soft body deformation, they struggle to simulate more complex soft body behaviors like cutting and suturing. A key challenge lies in modeling soft body fracture and splitting using the finite-element method (FEM), which is the predominant approach in current platforms. Additionally, the two-way suture needle/thread contact inside a soft body is further complicated when using FEM. In this work, we use the material point method (MPM) for such challenging simulations and propose new rigid geometries and soft-rigid contact methods specifically designed for them. We introduce CRESSim-MPM, a GPU-accelerated MPM library that integrates multiple MPM solvers and incorporates surgical geometries for cutting and suturing, serving as a specialized physics engine for surgical applications. It is further integrated into Unity, requiring minimal modifications to existing projects for soft body simulation. We demonstrate the simulator's capabilities in real-time simulation of cutting and suturing on soft tissue and provide an initial performance evaluation of different MPM solvers when simulating varying numbers of particles. The source code is available at https://github.com/yafei-ou/CRESSim-MPM.