  We evaluate the $e^- e^+ \to e^- e^+ +h$ process through the $ZZ$ fusion
channel at the International Linear Collider (ILC) operating at $500$ GeV and
$1$ TeV center of mass energies. We perform realistic simulations on the signal
process and background processes. With judicious kinematic cuts, we find that
the inclusive cross section can be measured to $2.9\%$ after combining the
$500$ GeV at $500 \text{fb}^{-1}$ and $1$ TeV at $1~ \text{ab}^{-1}$ runs. A
multivariate log-likelihood analysis further improves the precision of the
cross section measurement to $2.3\%$. We discuss the overall improvement to
model-independent Higgs width and coupling determinations and demonstrate the
use of different channels in distinguishing new physics effects in Higgs
physics. Our study demonstrates the importance of the $ZZ$ fusion channel to
Higgs precision physics, which has often been neglected in the literature.
