This paper investigates the robust stabilization problem for a class of neutral uncertain stochastic systems with Markovian switching and time-varying delays. The objective is to design a feedback controller based on discrete-time state observations and continuously available mode information such that the closed-loop system achieves robust exponential stability in the mean-square sense against all admissible uncertainties. By employing the Razumikhin technique, we derive sufficient conditions for mean-square exponential stability of the controlled system without imposing any differentiability assumption on the time-varying delay—only boundedness is required. This significantly relaxes the commonly used conditions in the existing literature, where the delay is required to be differentiable with a derivative bounded by a constant strictly less than one, or to satisfy a measure-type condition for non-differentiable delays. On this basis, the desired discrete-time feedback controller is explicitly constructed via linear matrix inequalities (LMIs), and an explicit estimate of the corresponding Lyapunov exponent is provided, which is rarely addressed in existing discrete-time feedback control results for neutral systems. Finally, numerical examples illustrate the effectiveness of the controller and demonstrate the applicability of the proposed criterion to a sampling-holding delay that violates commonly used differentiability and measure-type delay conditions.