Propagation of surface waves in a radially inhomogeneous chiral waveguide is considered. The setting is reduced to a boundary eigenvalue problem for the longitudinal components of the electromagnetic field in Sobolev spaces. To find the solution, variational formulation is used. The variational problem is reduced to the analysis of an operator-valued function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-valued function on the complex plane is determined. The results of numerical modeling of the spectrum of propagating surface waves in an open chiral waveguide are presented.