Criteria for Lyapunov stability of systems of nonlinear ordinary differential equations are developed under conditions of existence and uniqueness of the solution, continuity, and continuous differentiability of the system’s right-hand side. The criteria are constructed based on recurrent transformations of difference schemes and the finite difference formula. As a result of the transformations, criteria in the form of necessary and sufficient conditions are obtained. The form of the criteria allows for computer implementation, which leads to the possibility of real-time stability analysis. To compute with high accuracy the components of the solution and the right-hand side of the system included in the construction of the criteria, a piecewise interpolation method with iterative refinement is used. The criteria obtained using the finite difference formula are implemented with a parallelizable stable address sorting by merging. Examples of stability analysis of solutions of nonlinear systems based on the obtained criteria are presented. The analysis reduces to a numerical assessment of the expression from the left side of the criteria. The stability criteria differ from the known ones in that they do not require transformation of the right-hand side of the system and construction of the Lyapunov function. Based on the results of the program implementation of the criteria, a definitive conclusion can be made about the stability of the solutions of the studied system.