  We prove stability of integrable ALE manifolds with a parallel spinor under
Ricci flow, given an initial metric which is close in $L^p \cap L^\infty$, for
any $p \in (1, n)$, where $n$ is the dimension of the manifold. In particular,
our result applies to all known examples of $4$-dimensional gravitational
instantons. Our decay rates are strong enough to prove positive scalar
curvature rigidity in $L^p$, for each $p \in \left[1, \frac{n}{n-2}\right)$,
generalizing a result by Appleton.
