Despite recent advances in semantic Simultaneous Localization and Mapping (SLAM) for terrestrial and aerial applications, underwater semantic SLAM remains an open and largely unaddressed research problem due to the unique sensing modalities and the object classes found underwater. This paper presents an object-based semantic SLAM method for underwater environments that can identify, localize, classify, and map a wide variety of marine objects without a priori knowledge of the object classes present in the scene. The method performs unsupervised object segmentation and object-level feature aggregation, and then uses opti-acoustic sensor fusion for object localization. Probabilistic data association is used to determine observation to landmark correspondences. Given such correspondences, the method then jointly optimizes landmark and vehicle position estimates. Indoor and outdoor underwater datasets with a wide variety of objects and challenging acoustic and lighting conditions are collected for evaluation and made publicly available. Quantitative and qualitative results show the proposed method achieves reduced trajectory error compared to baseline methods, and is able to obtain comparable map accuracy to a baseline closed-set method that requires hand-labeled data of all objects in the scene.