The exceptional concentration of vertebrate diversity in continental freshwaters has been termed the “freshwater fish paradox,” with > 15,000 fish species representing more than 20% of all vertebrate species compressed into tiny fractions of the Earth’s land surface area (<0.5%) or total aquatic habitat volume (<0.001%). This study asks if the fish species richness of the world’s river basins is explainable in terms of river captures using topographic metrics as proxies. The River Capture Hypothesis posits that drainage-network rearrangements have accelerated biotic diversification through their combined effects on dispersal, speciation, and extinction. Yet rates of river capture are poorly constrained at the basin scale worldwide. Here we assess correlations between fish species density (data for 14,953 obligate freshwater fish species) and basin-wide metrics of landscape evolution (data for 3,119 river basins), including: topography (elevation, average relief, slope, drainage area) and climate (average rainfall and air temperature). We assess the results in the context of both static landscapes (e.g., species-area and habitat heterogeneity relationships) and transient landscapes (e.g., river capture, tectonic activity, landscape disequilibrium). We also relax assumptions of functional neutrality of basins (tropical vs. extratropical, tectonically stable vs. active terrains). We found a disproportionate number of freshwater species in large, lowland river basins of tropical South America, Africa, and Southeast Asia, under predictable conditions of large geographic area, tropical climate, low topographic relief, and high habitat volume (i.e., high rainfall rates). However, our results show that these conditions are only necessary, but not fully sufficient, to explain the basins with the highest diversity. Basins with highest diversity are all located on tectonically stable regions, places where river capture is predicted to be most conducive to the formation of high fish species richness over evolutionary timescales. Our results are consistent with predictions of several landscape evolution models, including the River Capture Hypothesis, Mega Capture Hypothesis, and Intermediate Capture Rate Hypothesis, and support conclusions of numerical modeling studies indicating landscape transience as a mechanistic driver of net diversification in riverine and riparian organisms with widespread continental distributions.