  Currently, the discussions and investigations for the vacuum energy is
drawing great both theoretical and experimental attention. The vacuum states of
variety of fields, subject to special boundary conditions, may contribute to
non-trivial macroscopic vacuum energy, i.e. the Casimir effect, which become an
interdisciplinary subject and plays an important role in a variety of fields of
physics. We adopt Schwinger's source theory and study the Casimir effect due to
the quantization of gravitation, i.e. the gravitational Casimir effect (GCE),
in inspiraling neutron star (NS) binaries with wide separation of $10^9 \rm m$.
By considering gravitoelectromagnetism (GEM) arising from the spiral-in orbital
motion and evaluating the contributions of GEM to the vacuum energy of
gravitons radiated during the orbital decay, we demonstrate that, when the
orbital separation of the binary decay a distance of $L$ in radial direction,
the GEM results in a small Casimir correction to the gravitational vacuum
energy, which contributes to an attractive gravitational Casimir force to the
binary, in addition the gravitational force. The gravitational waves (GWs),
emitted from wide inspiraling NS binaries, locate in the low-frequency band of
$10^{-4}-1$ Hz. For a characteristic GW frequency of $10^{-3}$ Hz, the
gravitational Casimir correction to the signals is estimated as of the order of
$\sim10^{-24}$, which corresponds to a force of $10^{-20}$ N. By considering
that the sensitivity of space-based gravitational wave observatory, LISA/eLISA
and Taiji, can be reduced to $10^{-24}$, we would expect that LISA/eLISA and
Taiji with sensitivity improvements give the powerful tool to detect GCE in the
near future.
