In this paper, we investigate a class of $\psi$-Caputo fractional differential hemivariational inequalities with history-dependent operators. By employing the Rothe method in conjunction with surjectivity results for multivalued pseudomonotone operators, the solvability of weak solutions to $\psi$-Caputo fractional differential hemivariational inequalities is obtained. As an application, a class of history-dependent viscoelastic friction contact problems that account for adhesion phenomena is investigated. In this contact model, the friction and contact conditions are described by Clarke generalized gradient of two nonconvex and nonsmooth function involving adhesion. Finally, the solvability of the solution for this friction contact model is established.