In this paper, we study the numerical approximation for the simplified Ericksen-Leslie system with variable density. Firstly, by denoting the orientation field vector d in the polar coordinate, we rewrite the Ericksen-Leslie system into an equivalent system, which allows the sphere constraint |d| = 1 to still be preserved at the discrete level. Secondly, we construct first-order and second-order time discretization schemes based on the convective form of the Gauge-Uzawa method. And we prove their unconditional energy stability. Then, through rigorous error analysis, we provide optimal convergence orders for all unknowns. We also construct a fully discrete scheme with finite elements in space and provide its stability result. Finally, some numerical simulations are provided to demonstrate the accuracy and effectiveness of the proposed schemes.