A modular adaptive controller is proposed for the nonlinear attitude tracking of micro-spacecraft in the presence of uncertain inertial tensor, external disturbances and saturation constraints, which combines the theory of system immersion and manifold invariance (I&I) and constrained backstepping. Firstly, the spacecraft attitude is globally represented by modified Rodrigues parameters. Then the backstepping controller is derived to implement any operating constraints and obtain the derivatives of virtual control easily by introducing command filters and modified tracking errors. To improve the robustness of baseline controller, the nonlinear estimator is constructed based on I&I to estimate and compensate the time-varying total disturbances on line. Because the I&I approach allows for prescribed uniformly stable dynamics to be assigned to the estimation error, the resulting modular adaptive controller is easier to tune compared to classical adaptive backstepping. And its performance does not suffer from unpredictable dynamical behavior of the estimation laws. The closed-loop input-to-state stability and boundedness of the estimation error are guaranteed by Lyapunov direct method. Comparative simulations state that the proposed controller is successful in achieving high attitude performance and precise estimation of disturbance.