The attitude tracking control problem is investigated for a rigid spacecraft with constrained inputs and model uncertainties. A bounded robust adaptive state feedback control law is developed based on a kind of bounded nonlinear continuous function and the bounded property of the modified Rodrigues parameters. The proposed control designing method gives the quantitative relationship between the constraint value, the bound of the desired trajectory, the bound of the model uncertainties, and the tunable controller gains. It is proved via the Lyapunov's stability theory that angular velocity errors tend to zero asymptotically and attitude errors converge to the small neighborhoods of zero by setting suitable controller parameters. The simulation results demonstrate that the proposed approach can achieve the design objective and deal with the model uncertainties under the constrained control inputs.