This paper is concerned with the robust gain-scheduling output feedback control problem for a class of linear parametervarying systems with time-varying state delay. The controlled plant under consideration is described as a linear fractional transformation (LFT) model of scheduling parameters. Dynamic integral quadratics (IQCs) are employed to characterize the input-output behavior of the state-delay nonlinearity. The robust stability and the L2-gain performance are first analyzed using quadratic Lyapunov function. Then, the design of dynamic output-feedback controllers robust against the plant state-delay nonlinearity and gain-scheduled by parameters is examined. The synthesis conditions of such robust gain-scheduling controllers are formulated in terms of linear matrix inequalities (LMIs) plus a line search, which can be solved effectively using existing algorithms. A numerical example has been used to demonstrate the effectiveness and advantages of the proposed approach.