The rigid-frame bridges with a fully continuous moment connection between the superstructure and substructure piers are suitable for high-seismic regions; however, research on the response of these bridge types under tsunami loads is limited. This paper presents a framework for developing fragility functions of coastal rigid-frame bridges, and bridges with strong pier-to-deck connectivity, undergoing tsunami-induced hydrodynamic loads. The fragility analysis is demonstrated on a three-span reinforced concrete (RC) bridge structure with double-column bents. The Monte Carlo simulation is implemented to consider the uncertainties in capacity and demand parameters. The efficiency of single-parameter and two-parameter intensity measures for predicting the response of bridges is examined. The viability of strengthening bridge piers using fiber-reinforced polymer (FRP) is also investigated. The results indicate that the use of two-parameter intensity measures is a better option to estimate the structural response since the hydrodynamic forces are a function of both the flow depth and velocity. This also leads to a significant reduction in the scatter of fragility data. The comparative assessment of the initial and FRP-strengthened bridges highlights that the FRP-jacketing is an effective method in reducing the risk of failure of the rigid-frame bridges.