No, you had better not.
The culprit of conjunctivitis is a bacterium called amoeba which exist in tap water.
Also, tap water or other water can't clean your contact lenses.
The main purpose of rinsing contact lenses is to sterilize and deproteinize them. Only contact solutions can do so.
No, you can not rinse your contact lenses with water which may contain a lot of invisible bacterium. You should better buy the new bottle of contact solution. Or else, your contact lenses will get the bacterium which will make you get the eye infection next time when you wear. You should keep this in mind. The way of wearing the contact lenses is so important which you must keep the good hygiene.
So, it seems that you have a shortage of solution. As a matter of fact, tap water would clean your contact lenses, and reduce those unnecessary protein. However, I should remind you tap water might get some bacteria on your lenses, giving rise to some infections. So you should try not to use water in the long run. Just take care of your eyes.