We consider an energy harvesting sensor transmit- ting latency-sensitive data over a fading channel. We aim to find the optimal transmission scheduling policy that minimizes the packet queuing delay given the available harvested energy. We formulate the problem as a Markov decision process (MDP) over a state-space spanned by the transmitter's buffer, battery, and channel states, and analyze the structural properties of the resulting optimal value function, which quantifies the long-run performance of the optimal scheduling policy. We show that the optimal value function (i) is non- decreasing and has increasing differences in the queue backlog; (ii) is non-increasing and has increasing differences in the battery state; and (iii) is submodular in the buffer and battery states. Our numerical results confirm these properties and demonstrate that the optimal scheduling policy outperforms a so-called greedy policy in terms of sensor outages, buffer overflows, energy efficiency, and queuing delay.