  Let $(X,\omega)$ be a compact Hermitian manifold. We establish a stability
result for solutions to complex Monge-Amp\`ere equations with right-hand side
in $L^p$, $p>1$. Using this we prove that the solutions are H\"older continuous
with the same exponent as in the K\"ahler case \cite{DDGKPZ14}. Our techniques
also apply to the setting of big cohomology classes on compact K\"ahler
manifolds.
