River drainage networks are important landscape features that have been studied for several decades from a range of geomorphological and hydrological perspectives. However, identifying the most vital (critical) nodes on river networks and analyzing their relationships with geomorphic and climatic properties have not yet been extensively addressed in the literature. In this study, we use an algorithm that determines the set of critical nodes whose removal results in maximum network fragmentation and apply it to various topologies of simulated and natural river networks. Specifically, we consider simulated river networks obtained from optimal channel network (OCN) approach as well as extracted networks from several natural basins across the United States. Our results indicate a power-law relationship between the number of connected node pairs in the remaining network and the number of removed critical nodes. We also investigate the characteristics of sub-basins resulted from the removal of critical nodes and compare them with those of central nodes (in the context of betweenness centrality) for both natural basins and OCNs with varying energy exponent γ to understand vulnerability and resilience of river networks under potential external disruptions.