Stability of the traveling wave solution to a general class of one-dimensional nonlocal evolution equations is studied in $L^2$-spaces, thereby providing an alternative approach to the usual spectral analysis with respect to the supremum norm. We prove that the linearization around the traveling wave solution satisfies a Lyapunov-type stability condition in a weighted space $L^2(ρ)$ for a naturally associated density $ρ$. The result can be applied to obtain stability of the traveling wave solution under stochastic perturbations of additive or multiplicative type. For small wave speeds, we also prove an alternative Lyapunov-type stability condition in $L^2(\mathfrak{m})$, where $\mathfrak{m}$ is the symmetrizing density for the traveling wave operator, which allows to derive a long-term stochastic stability result.