Leave the speaker driver alone. You will do more harm than good trying to remove the urine. Dog urine is has a significant mineral load. This will change the Fs of that driver.
If you want to preserve your stereo image, have him urinate on the contralateral driver--all will be good.
(Sorry man--couldn't resist. Seriously, you will probably never hear the urine burden but if it drives you nuts you can recone or replace the driver.)