Can I trigger the RF-603 (and thus, flash), via PC/other cable?

No. You can't. The PC sync port on the RF-603 is output-only for receiver mode to connect to the flash, not an input. I know, because I tested it with an optical sensor that worked just fine on my very very old RF-602-TX unit (no longer sold as part of the RF-602 sets--they removed the PC port).

There are, however, radio triggers that do have sync inputs as well as outputs. Godox triggers, for example, typically have sync ports that can be configured to be either input or output. The Godox X-Pro transmitter, for example, has a 2.5mm sync port, and the SYNC C.Fn can be set to IN or OUT.

Or, if you don't mind going corded, just sending the output from the acoustic sensor directly to the flash via its PC port over a cable would work. If you're having a hard time sourcing PC connectors/cables, you can also consider modding the YN-560III to add a 3.5 miniphone connector (or just buying a flash with a 3.5mm jack, like a LumoPro LP 120/160/180), or use a hotshoe-to-3.5mm adapter. The 3.5mm miniplugs are more robust than PC connectors, anyway.

If you want to trigger the RF-603, I'd recommend getting a hot shoe adapter like the ones inkista mentioned at the end of her answer. There are various brands and models that accept all sorts of connections (pc port, 3.5 mm stereo, 1/4" stereo, etc.), and it should be easy to find one that matches whatever your acoustic trigger provides. They're simple devices and only cost $10 or so. You would wire your acoustic trigger to the port on the hot shoe and mount the radio trigger on the hot shoe. When the acoustic trigger fires, it will transmit that signal to the hot shoe adapter and fire the radio trigger.

Note that you'll introduce some delay when triggering the flash with a radio trigger instead of directly wiring to the flash. Then again, using a radio trigger makes it simple to expand your setup to trigger multiple lights.

If the transmitter can be triggered via hotshoe, any passive adapter that has a female hotshoe and a PC sync port connected to it could be used. Such adapters exist (eg Vello HSA-PSU), sometimes in a variation that has a male hotshoe on the bottom, female hotshoe on top, and a PC port - although they are not mainstream, given the potential for damaging flashes when used incorrectly (eg attempting to connect two dissimilar flashguns in parallel).

I use a Pluto Trigger for waterdrop photography .... Pluto trigger is plugged into the shutter release on the camera and to the water reservoir valve.... 1 RF603 on the camera hotshoe and another RF603 on a SB700 speedlight. works great