Aligning with the global mandates pushing towards advanced technologies with reduced resource consumption and environmental impacts, the sustainability of wireless networks becomes a significant concern in 6G systems. To address this concern, a native integration of sustainability into the operations of next-generation networks through novel designs and metrics is necessary. Nevertheless, existing wireless sustainability efforts remain limited to energy-efficient network designs which fail to capture the environmental impact of such systems. In this paper, a novel sustainability metric is proposed that captures emissions per bit, providing a rigorous measure of the environmental footprint associated with energy consumption in 6G networks. This metric also captures how energy, computing, and communication resource parameters influence the reduction of emissions per bit. Then, the problem of allocating the energy, computing and communication resources is posed as a multi-objective (MO) optimization problem. To solve the resulting non-convex problem, our framework leverages MO reinforcement learning (MORL) to maximize the novel sustainability metric alongside minimizing energy consumption and average delays in successfully delivering the data, all while adhering to constraints on energy resource capacity. The proposed MORL methodology computes a global policy that achieves a Pareto-optimal tradeoff among multiple objectives, thereby balancing environmental sustainability with network performance. Simulation results show that the proposed approach reduces the average emissions per bit by around 26% compared to state-of-the-art methods that do not explicitly integrate carbon emissions into their control objectives.