To fully realize the potential of vehicular networks, several obstacles and challenges need to be addressed. Chief among the obstacles are strict QoS requirements of applications and differentiated service requirements in different situations. Although DSRC and WAVE have been adopted as the de facto standards, they do not address all the problems and there is room for improvements. In this study, we propose a generic prioritization and resource management algorithm that can be used to prioritize processing of received packets in vehicular networks. We formulate the generic severity-based prioritized packet processing problem as Penalized Multiple Knapsack Problem (PMKP) and prove that it is an NP-Hard problem. We thus develop a real-time heuristic that utilizes a relaxed version of the formulation. The relaxed formulation executes in polynomial time and guarantees a minimum delay per severity-level while respecting the processing rate constraint. To measure the performance of the proposed heuristic, real traffic data is used in a small-scale experiment. The proposed heuristic is tested against the PMKP solution and results show a small degradation of up to 4 percent in profit for the heuristic compared to the PMKP solution. Also, the proposed heuristic is tested against a non-prioritized processing algorithm that works using first come first served policy. Results show that the proposed heuristic gains 9 to 67 percent more profit than the non-prioritized processing algorithm in moderate and high congestion scenarios.