Asymptotic distribution of the discrete spectrum eigenvalues for a recently developed model of a double-walled carbon nanotube is presented in this paper. The corresponding initial boundary-value problem is reduced to an evolution equation, whose dynamics generator is a non-self-adjoint matrix differential operator with a purely discrete spectrum. It is shown in the paper that the entire spectrum asymptotically splits into four spectral branches. Asymptotic representation is derived for the eigenvalues along each spectral branch as the number of an eigenvalue tends to infinity. To prove the results, a two-step procedure involving the construction of the left- and right-reflection matrices has been used.