A Moyal deformation of a Clifford $ Cl (3, 1) $ Gauge Theory of (Conformal) Gravity is performed for canonical noncommutativity (constant $\Theta^{\mu \nu }$ parameters). In the very special case when one imposes certain constraints on the fields, there are $no$ first order contributions in the $\Theta^{\mu \nu }$ parameters to the Moyal deformations of Clifford gauge theories of gravity. However, when one does $not$ impose constraints on the fields, there are first order contributions in $\Theta^{\mu \nu }$ to the Moyal deformations in variance with the previous results obtained by other authors and based on different gauge groups. Despite that the generators of $U(2,2), SO(4,2), SO(2,3)$ can be expressed in terms of the Clifford algebra generators this does $not$ imply that these algebras are isomorphic to the Clifford algebra. Therefore one should not expect identical results to those obtained by other authors. In particular, there are Moyal deformations of the Einstein-Hilbert gravitational action with a cosmological constant to first order in $\Theta^{\mu \nu }$ . Finally, we provide a mechanism which furnishes a plausible cancellation of the huge vacuum energy density.