Fixing Water Damaged iPhone
Before you continue, make sure you read all of the links I have posted in this answer below. The most important link is this one, which indicates that:

If an iPhone or iPod has been damaged by liquid (for example, coffee
or a soft drink), the service for such liquid damage is not covered by
the Apple one year limited warranty or an AppleCare Protection Plan
(APP). iPhone and most iPod devices that were built after 2006 have
built-in Liquid Contact Indicators that will show (as described below)
whether the device has been in contact with water or a liquid
containing water.


The Solution Everyone Does
Everyone that I hear who drops their phone into water (either submerged into the toilet or onto their screen) does something like this:

I immediately but the phone in a bag of rice for 4 days. When I took
it out I tried to turn it on but nothing. Then I tried plugging it
into my computer and nothing happened as well. I left the phone plugged
into the computer for about half an hour then all of a sudden the
computer recognized my phone. I then plugged my phone into an outlet
and it made the charging sound. I thought my phone was going to be
okay but then suddenly I realized that my screen was not coming on.
You have to turn the phone at a certain angle to see a part of
the screen. The strange part is the touchscreen still works.

Bottom line is, there is a 95% chance you won't be able to fix it. But try this, and you may be able to get it working at least partially. Make sure you take it an Apple Store first for advice. They may be able to let you replace it or at least charge a (somewhat) small fee.
Other Important Information Regarding Water Sensors
Apple has water damage sensors in their phone and if it is completely submerged, they will not replace your phone (probably). Submerging it into alcohol (see below) WILL trigger these sensors. Also, if your phone was completely submerged, the sensors would have already triggered. See this for complete information.
Risky Solutions
I'm going to list some solutions you can try (at your own risk). These are from people who have had similar issues as yours.

The water probably has left some minerals behind that is stuck in
crevices, causing the instability. You'll want to wash out the
residue. Get isopropyl alcohol, nothing below 95%, if you can even get
99% it would be best. Simply set your device into it completely
submerged for 15 minutes. Set back into rice for a night or two. The
alcohol will wash and push out the minerals left behind and the rice
will facilitate by absorbing it out.

Source: iPhone Water Damaged Screen