Underwater optical wireless communications (UOWC) is an emerging technology to provide underwater applications with high speed and low latency connections. However, it suffers from limited range and requires effective multi-hop routing solutions for the proper operation of underwater optical wireless networks (UOWNs). In this regard, this paper proposes a distributed Sector-based Opportunistic Routing (SectOR) protocol. Unlike the traditional routing techniques which unicast packets to a unique relay, opportunistic routing (OR) targets a set of candidate relays by leveraging the broadcast nature of the UOWC channel. OR is especially suitable for UOWNs as the link connectivity can be disrupted easily due to the underwater channel impairments (e.g., pointing errors, misalignment, turbulence, etc.) and sea creatures passing through the transceivers' line-of-sight. In such cases, OR improves the packet delivery ratio as the likelihood of having at least one successful packet reception is much higher than that in conventional unicast routing. Contingent upon the performance characterization of a single-hop link, we obtain distance progress (DP) and expected (DP) metrics to evaluate the fitness of a candidate set (CS) and prioritize the members of a CS. Since rate↔error and range↔beamwidth tradeoffs yield different candidate set diversities, we develop a candidate selection and prioritization (CSPA) algorithm to find the optimal sector shaped coverage region by scanning the feasible search space. Moreover, a hybrid acoustic/optic coordination mechanism is considered to avoid duplicate transmission of the relays. Numerical results show that SectOR protocol can perform even better than an optimal unicast routing protocol in well-connected UOWNs.