This paper considers a heterogeneous ad hoc network with multiple transmitter-receiver pairs, in which all transmitters are capable of harvesting renewable energy from the environment and compete for one shared channel by random access. In particular, we focus on two different scenarios: the constant energy harvesting (EH) rate model where the EH rate remains constant within the time of interest and the i.i.d. EH rate model where the EH rates are independent and identically distributed across different contention slots. To quantify the roles of both the energy state information (ESI) and the channel state information (CSI), a distributed opportunistic scheduling (DOS) framework with two-stage probing and save-then-transmit energy utilization is proposed. Then, the optimal throughput and the optimal scheduling strategy are obtained via one-dimension search, i.e., an iterative algorithm consisting of the following two steps in each iteration: First, assuming that the stored energy level at each transmitter is stationary with a given distribution, the expected throughput maximization problem is formulated as an optimal stopping problem, whose solution is proved to exist and then derived for both models; second, for a fixed stopping rule, the energy level at each transmitter is shown to be stationary and an efficient iterative algorithm is proposed to compute its steady-state distribution. Finally, we validate our analysis by numerical results and quantify the throughput gain compared with the best-effort delivery scheme.