  We prove that minimal instanton bundles on a Fano threefold $X$ of Picard
rank one and index two are semistable objects in the Kuznetsov component
$\mathsf{Ku}(X)$, with respect to the stability conditions constructed by
Bayer, Lahoz, Macr\`i and Stellari. When the degree of $X$ is at least $3$, we
show torsion free generalizations of minimal instantons are also semistable
objects. As a result, we describe the moduli space of semistable objects with
same numerical classes as minimal instantons in $\mathsf{Ku}(X)$. We also
investigate the stability of acyclic extensions of non-minimal instantons.
