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The demand for same-day delivery (SDD) has increased rapidly in the last few years and has particularly boomed during the COVID-19 pandemic. The fast growth is not without its challenge. In 2016, due to low concentrations of memberships and far distance from the depot, certain minority neighborhoods were excluded from receiving Amazon's SDD service, raising concerns about fairness. In this paper, we study the problem of offering fair SDD-service to customers. The service area is partitioned into different regions. Over the course of a day, customers request for SDD service, and the timing of requests and delivery locations are not known in advance. The dispatcher dynamically assigns vehicles to make deliveries to accepted customers before their delivery deadline. In addition to the overall service rate (utility), we maximize the minimal regional service rate across all regions (fairness). We model the problem as a multi-objective Markov decision process and develop a deep Q-learning solution approach. We introduce a novel transformation of learning from rates to actual services, which creates a stable and efficient learning process. Computational results demonstrate the effectiveness of our approach in alleviating unfairness both spatially and temporally in different customer geographies. We also show this effectiveness is valid with different depot locations, providing businesses with an opportunity to achieve better fairness from any location. Further, we consider the impact of ignoring fairness in service, and results show that our policies eventually outperform the utility-driven baseline when customers have a high expectation on service level.