This paper deals with the problem of robust output regulation of systems governed by nonlinear contraction semigroups. After adding an integral action to the system, we design a feedback law based on the so-called forwarding approach. For small constant perturbations, we give sufficient conditions for the existence of a locally exponentially stable equilibrium at which the output coincides with the reference. Under additional assumptions, global asymptotic stability is achieved. All these conditions are investigated in the case of semilinear systems, and examples of application are given.