Internet of underwater things (IoUT) is a technological revolution that could mark a new era for scientific, industrial, and military underwater applications. To mitigate the hostile underwater channel characteristics, this paper hybridizes underwater acoustic and optical wireless communications to achieve a ubiquitous control and high-speed low-latency networking performance, respectively. Since underwater optical wireless communications (UOWC) suffers from limited range, it requires effective multi-hop routing solutions. In this regard, we propose a Sector-based Opportunistic Routing (SectOR) protocol. Unlike the traditional routing (TR) 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 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 a variety of local and global 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 filtering and searching 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.