This paper presents a novel rigid-body spacecraft navigation and control architecture within the framework of special Euclidean group SE(3) and its tangent bundle TSE(3) while considering stochastic processes in the system. The proposed framework combines the orbit-attitude motions of the spacecraft into a single, compact set. The stochastic state filter is designed based on the unscented Kalman filter which uses a special retraction function to encode the sigma points onto the manifold. The navigation system is then integrated to an almost globally asymptotically stabilizing Morse-Lyapunov-based control system with backstepping. Numerical simulations are conducted to demonstrate the effectiveness of the proposed navigation filter for the full state estimation. In addition, the navigation and control system is tested in the nonlinear gravity field of a small celestial body with an irregular shape. In particular, the performance of the closed-loop system is studied in a tracking problem of spacecraft motion near the asteroid Bennu based on the OSIRIS-REx's mission data.