We present a new approximation to include fully general relativistic pressure and velocity in Newtonian hydrodynamics. The energy conservation, momentum conservation and two Poisson's equations are consistently derived from Einstein's gravity in the zero-shear gauge assuming weak gravity and action-at-a-distance limit. The equations show proper special relativity limit in the absence of gravity. Our approximation is complementary to the post-Newtonian approximation and the equations are valid in fully nonlinear situations.PACS numbers: 04.25.Nx, 95.30.Lz, 95.30.Sf 1. Introduction: Considering the enormous practical and conceptual difficulties in handling general relativistic astrophysical situations using numerical simulations of full Einstein's gravity [1], it is always welcome to have an approximation method. The post-Newtonian (PN) approximation is one such method [2-5] where we restore the good and old absolute space and absolute time, and regard Einstein's gravity effects as corrections to the Newtonian equations. In this way we can handle weak but relativistic effects of gravity in Newtonian style, and the resulting equations are fully nonlinear.Here, we provide a complementary approximation which can handle the fully relativistic pressure and velocity in the weak gravity and action-at-a-distance limit: for our assumptions see Eq. (7). We present the energy conservation, momentum conservation and Poisson's equation which allow us to handle such astrophysical situations in Newtonian manner. In this approximation also the equations are valid to fully nonlinear orders. Our derivation is based on the zero-shear gauge which will be explained later. We ignore the transverse-tracefree tensor-type perturbation in the spatial metric, and ignore the anisotropic stress.2. Result: A closed form of new hydrodynamic equations we are proposing is