The paper develops a real-time vehicle-specific eco-routing model for use in on-board navigation systems for both conventional fuel- and battery-powered electric vehicles. The model uses a novel link cost function that retains all microscopic transient behavior along a link by capturing within-link speed, acceleration, and road grade variations. Specifically, eight vehicle-agnostic link-specific variables are transmitted to the cloud and fused with information received from other connected vehicles. The fused data are then sent back to the connected vehicles and used to compute vehicle-specific eco-routes (edge computing). The solution produces an energy-efficient dynamic user-equilibrium traffic assignment that is solved incrementally. A numerical experiment is first designed to test the model, demonstrating the benefits associated with vehicle-specific eco-routes compared to those produced by traditional models. The model is then implemented in a micro-simulation framework to evaluate the network-wide impacts of the proposed energy-efficient user-optimum incremental traffic assignment for different traffic demand levels. Results demonstrate that the proposed model produces lower network-wide energy consumption levels compared to traditional energy-efficient and travel time user-optimum traffic assignment methods for both conventional fuel and electric vehicles. Conventional fuel savings relative to travel time routing decrease with increasing traffic demand levels, while electric power savings do not vary monotonically with congestion levels. Energy savings relative to traditional eco-routing are also demonstrated to not vary monotonically with demand levels. Results of the study also demonstrate that the transportation network layout significantly affects the eco-routing system performance. Finally, the feasibility of the proposed model in real-world applications is discussed and presented.