We develop an immersed boundary projection method for solving the Naiver-Stokes equations and Newton-Euler equations to simulate the fluid-rigid body interactions in two and three dimensions. A novel fractional step algorithm is introduced for which fast solvers can be applied by exploiting the algebraic structure of the underlying schemes. The Navier-Stokes equations are decoupled while the Newton-Euler equations are solved simultaneously with a constraint equation of the immersed boundary force density. In contrast to previous works, the present method preserves both the fluid incompressibility and the kinematic constraint of the rigid body dynamics at a discrete level simultaneously while maintaining numerical stability. We demonstrate the numerical results of the present method involving spherical and spheroidal rigid bodies with a moderate range of density ratios, which are congruent with the results in the literature.