The focus of this study was to investigate a discrete-time stochastic predator-prey model. This model included environmental noise, temporal delay, and a nonlinear Ivlev-type functional response. First of all, we proved boundedness, positivity, first, and second moment boundedness, characteristic equation, and Jury conditions. Further, using the characteristic equation and Jury conditions, we determined equilibrium, exponential mean-square stability, and local stability which prevented the system from becoming unstable as a result of random shocks. Additionally, we studied fractional memory effects on the system and demonstrated how the delay parameter exhibited Neimark–Sacker (discrete Hopf) bifurcation. The graphical analysis and the numerical simulations validated the theoretical results.