In this work, we propose Runge-Kutta time integration schemes for the incompressible Navier-Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. The segregated Runge-Kutta methods are motivated as an implicit-explicit Runge-Kutta time integration of the projected Navier-Stokes system onto the discrete divergence-free space, and its re-statement in a velocity-pressure setting using a discrete pressure Poisson equation. We have analysed the preservation of the discrete divergence constraint for segregated Runge-Kutta methods and their relation (in their fully explicit version) with existing half-explicit methods. We have performed a detailed numerical experimentation for a wide set of schemes (from first to third order), including implicit and IMEX integration of viscous and convective terms, for incompressible laminar and turbulent flows. Further, segregated Runge-Kutta schemes with adaptive time stepping are proposed.