  We prove gauge-independence of one-loop path integral for on-shell quantum
gravity obtained in a framework of modified geometric approach. We use
projector on pure gauge directions constructed via quadratic form of the
action. This enables us to formulate the proof entirely in terms of
determinants of non-degenerate elliptic operators without reference to any
renormalization procedure. The role of the conformal factor rotation in
achieving gauge-independence is discussed. Direct computations on $CP^2$ in a
general three-parameter background gauge are presented. We comment on gauge
dependence of previous results by Ichinose.
