Compared with conventional methods that require electric manipulation of the transducer, passive diffraction gratings provide a simpler and more compact solution to generate an acoustic vortex. This work theoretically and numerically investigates a novel method for generation and detection of an acoustic vortex in water by using a passive Fermat's spiral diffraction grating (FSDG). When the proposed structure is illuminated by a plane acoustic wave, the constructive and destructive interferences of the diffracted waves by the FSDG could produce acoustic vortex beams. By changing the structure of the FSDG properly, the output can be high-order acoustic vortex or multiplexing acoustic vortex beams, whose topological charge is determined by the number and spiral direction of the FSDG arms. Then, the acoustic vortex beams can be detected by the same equipment but a different rotation direction. Both the intensity and phase distributions after detection demonstrate the effectiveness of the designed FSDG for generating and detecting the acoustic vortex beams. Obtained results may have potential applications for particle trapping, imaging, underwater acoustic communication, and so on.