  Noncommutative gravity is a natural method of quantizing spacetime by
promoting the spacetime coordinates themselves to operators which do not
commute. This approach is motivated, for example, from a quantum gravity
perspective, among others. Noncommutative gravity has been tested against the
binary black hole merger event GW150914. Here, we extend and improve upon such
a previous analysis by (i) relaxing an assumption made on the preferred
direction due to noncommutativity, (ii) using posterior samples produced by the
LIGO/Virgo Collaborations, (iii) consider other gravitational wave events,
namely GW151226, GW170608, GW170814 and GW170817, and (iv) consider binary
pulsar observations. Using Kepler's law that contains the noncommutative effect
at second post-Newtonian order, we derive corrections to the gravitational
waveform phase and the pericenter precession. Using the gravitational wave and
double pulsar binary observations, we find bounds on a space-time
noncommutative tensor $\theta^{0i}$ in terms of the preferred frame direction
with respect to the orientation of each binary. We find that the gravitational
wave bounds are stronger than the binary pulsar one by an order of magnitude
and the noncommutative tensor normalized by the Planck length and time is
constrained to be of order unity.
