Deformable image registration is one of the essential processes in analyzing medical images. In particular, when diagnosing abdominal diseases such as hepatic cancer and lymphoma, multi-domain images scanned from different modalities or different imaging protocols are often used. However, they are not aligned due to scanning times, patient breathing, movement, etc. Although recent learning-based approaches can provide deformations in real-time with high performance, multi-domain abdominal image registration using deep learning is still challenging since the images in different domains have different characteristics such as image contrast and intensity ranges. To address this, this paper proposes a novel unsupervised multi-domain image registration framework using neural optimal transport, dubbed OTMorph. When moving and fixed volumes are given as input, a transport module of our proposed model learns the optimal transport plan to map data distributions from the moving to the fixed volumes and estimates a domain-transported volume. Subsequently, a registration module taking the transported volume can effectively estimate the deformation field, leading to deformation performance improvement. Experimental results on multi-domain image registration using multi-modality and multi-parametric abdominal medical images demonstrate that the proposed method provides superior deformable registration via the domain-transported image that alleviates the domain gap between the input images. Also, we attain the improvement even on out-of-distribution data, which indicates the superior generalizability of our model for the registration of various medical images. Our source code is available at https://github.com/boahK/OTMorph.