In licensed-assisted access using the LTE (LAA) standard, carrier sensing via the listen-before-talk (LBT) procedure is a vital feature for fair sharing with the Wi-Fi systems. Furthermore, it has been designed to support frequency reuse-1 operation among all cells by the virtue of licensed spectrum. As opposed to the two existing channel-access schemes for frequency reuse-1, transmission start time alignment (TSTA) and energy detection threshold adaptation (EDTA), which may not be able to maximize the LAA system throughput without violating the requirement of fair coexistence, we propose a new frequency-reuse-1 scheme, referred to as the alignment reference interval adaptation-based LAA (ARIA-LAA). It attempts to combine the advantages of TSTA and EDTA into a unified access framework, in which the alignment reference interval (ARI) is adaptively adjusted to control the channel-access probability for LAA and Wi-Fi systems. Meanwhile, to operate the ARIA-LAA effectively toward our design objective, we design the fuzzy Q-learning system that adapts the continuous variable ARI to the dynamically changing wireless network environment. Based on the analytical system models and formation of the optimization problem, it employs a model-free learning algorithm that interacts with the state, defined as the current level of fairness achieved by adaptation, and the Q-learning function to determine the ARI as a global action. Our simulation results demonstrate that the ARIA-LAA is a novel scheme of spectrum sharing with spatial reuse for LAA that enhances the overall system capacity while satisfying the fair-coexistence requirement in the unlicensed spectrum.