This paper generalizes Witten rigidity to establish rigidity results in equivariant \(K\)-theory for indices of families of \(S^{1}\)-invariant twisted \(\text{Spin}^{c}\) Dirac operators. On a closed spin manifold the Dirac operator can be ``twisted'' by a vector bundle. The resulting operator has a Fredholm index. When all structures are \(S^{1}\)-invariant, the index can be viewed as a character of \(S^{1}\). Rigidity is the statement that this character is a constant function on \(S^{1}\). The Witten rigidity theorem (see, e.g., \textit{E. Witten} [Commun. Math. Phys. 109, 525--536 (1987; Zbl 0625.57008)] and \textit{R. Bott} and \textit{C. Taubes} [J. Am. Math. Soc. 2, No. 1, 137--186 (1989; Zbl 0667.57009)].) identifies sequences of twisting vector bundles for which the twisted Dirac operator is rigid. Included as one case is a sequence of twists of the signature operator. For a family of operators of the type described above, the index is an element of the \(S^{1}\)-equivariant \(K\)-theory of the space parametrizing the family. A strong form of rigidity in this situation is equality of this index and its \(S^{1}\)-invariant part. For a family of signature operators arising from a fibration of closed manifolds with spin fibers, \textit{K. Liu}, \textit{X. Ma}, and \textit{W. Zhang} [C. R. Acad. Sci., Paris, Sér. I, Math. 330, No. 4, 301--305 (2000; Zbl 0951.55008)] identify a sequence of twisting bundles for which the resulting family of twisted signature operators is rigid. The paper under review extends this result to a family of \(\text{Spin}^{c}\) Dirac operators. Sources of examples include fibrations with almost complex fibers. In addition to rigidity results, this paper includes vanishing results for equivariant family indices. The paper's treatment of the \(\text{Spin}^{c}\) case requires the introduction of auxiliary vector bundles whose characteristic classes satisfy certain conditions. The proofs in this paper depend on a \(K\)-theoretic equivariant family index theorem whose analytic proof involves techniques introduced by S. Wu, W. Zhang, J.-M. Bismut, and G. Lebeau.