We obtain a manifestly background independent BRST quantization of the N = 4 supersymmetric spinning particle. We show that nilpotency of the BRST charge Q implies the Einstein equations admitting a cosmological constant of indefinite sign. The physical graviton states are given by the vertex operator, obtained by an infinitesimal variation of Q, acting on diffeomorphism ghost states. In addition, the tree-point graviton scattering vertex is correctly reproduced by the worldline computation with these vertex operators.