To address the degradation of control performance caused by significant changes in the gravitational environment during the transition of space manipulators from ground assembly and testing to on-orbit operation, this paper proposes a gravity-adaptive predefined-time trajectory tracking control method. The proposed method exploits the linear parameterization structure of the gravitational term in manipulator dynamics to design a scalar adaptive law for online estimation and real-time compensation of varying gravitational environments. By incorporating the parameter estimation error into the Lyapunov function, the gravity adaptation and predefined-time convergence are unified within a single theoretical framework. This enables a single set of controller parameters to be directly applied to both terrestrial gravitational and space microgravity environments without re-tuning. The inertial parameter perturbations induced by payload variations are treated as bounded disturbances within the robustness analysis. The stability and predefined-time convergence of the closed-loop system are rigorously established using Lyapunov theory. Simulation results validate the effectiveness of the proposed method under different gravitational environments and payload conditions. The proposed method provides a viable solution for unified control of space manipulators from ground assembly to on-orbit application.