  In this paper, we study the non commutative quantum dynamics of the spinorial
field in the presence of a gravitational background within DeWitt's approach.
By employing the vierbein formalism in the temporal state of non commutative
four-dimensional space time, we also analyze the first level correction of the
solution and Green's function in terms of the Riemann curvature tensor. The
effect of the gravitational background is imposed at least up to the first
order of the Riemann curvature tensor.
