Automated volumetric meshing of patient-specific heart geometry can help expedite various biomechanics studies, such as post-intervention stress estimation. Prior meshing techniques often neglect important modeling characteristics for successful downstream analyses, especially for thin structures like the valve leaflets. In this work, we present DeepCarve (Deep Cardiac Volumetric Mesh): a novel deformation-based deep learning method that automatically generates patient-specific volumetric meshes with high spatial accuracy and element quality. The main novelty in our method is the use of minimally sufficient surface mesh labels for precise spatial accuracy and the simultaneous optimization of isotropic and anisotropic deformation energies for volumetric mesh quality. Mesh generation takes only 0.13 seconds/scan during inference, and each mesh can be directly used for finite element analyses without any manual post-processing. Calcification meshes can also be subsequently incorporated for increased simulation accuracy. Numerous stent deployment simulations validate the viability of our approach for large-batch analyses. Our code is available at https://github.com/danpak94/Deep-Cardiac-Volumetric-Mesh .