This paper considers the problem of estimating the trajectory of an autonomous underwater vehicle (AUV) via a single passive receiver, without any anchor nodes or receiving arrays, and with the only help of a sequence of known acoustic signals emitted by the AUV. This scenario is of interest in case multilateration-based alternatives would require the deployment of many receivers and imply exceedingly high costs, e.g., for the coverage of wide areas. The proposed method exploits the knowledge of environmental parameters such as the sound speed profile, bathymetry and bottom sediments in order to estimate the location of the AUV, taking advantage of the spatial dependency of channel impulse responses that arises from the diverse bathymetry around the receiver. This dependency is captured by comparing channel estimates against a database of channel responses, pre-computed through an acoustic propagation model. This yields multiple likely AUV locations, which are filtered via a path tracking method similar to the Viterbi algorithm, in order to estimate the trajectory of the AUV. Results obtained both from simulations and from a sea experiment show that the proposed method can estimate node locations and paths with a small error, especially considering the use of a single receiver.