A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretizations. It is known that this coercivity property is inherited by convolution quadrature time discretization based on A-stable multistep methods, which are of order at most 2. Here we study the question as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge–Kutta methods and hence for methods of arbitrary order. As an illustration the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretization of a nonlinear boundary integral equation that originates from a nonlinear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretization.