By the time you posted this it was too late. The first thing you do when a phone gets wet is immediately take the battery out. You should have done that before getting up. In this case, immediately means 5 seconds ago.
The third worst thing you can do with a wet phone is leave the battery in. The second worst thing is turning it on. The worst? Charging it. The step after charging a wet phone is removing the cards, if any (SD and SIM) and crushing the phone to bits with a hammer, so no one can recover any sensitive data in it. Charging it turned it from a phone to a lump of partially melted silicon with some metal and plastic parts.
If the phone wasn't damaged by having the battery in, and further damaged by turning it on, get a non-metallic pan large enough to hold about a quart of alcohol, plain old 70% rubbing alcohol. A glass meat loaf pan is about ideal. Put the phone (the battery has been removed) in the alcohol and swish it around vigorously for about 5 minutes. You want to wash any minerals, dirt, impurities, etc., out into the alcohol. Don't worry about wetting the phone - alcohol isn't water, it absorbs water.
After about 5 minutes of this, drain the phone, dump the alcohol (it's got water from the phone in it), replace it with fresh alcohol and repeat for another 5 minutes. Then one more time. Three baths in all.
Wipe the battery down with alcohol on a paper towel. The battery is sealed, you just want to keep from putting any crap into the phone when you put the battery back in.
Put the battery and the phone into a container of UNCOOKED rice. Some people use a plastic bag, but I prefer a gallon jar or a LARGE plastic container that can be sealed. Make sure the phone is completely covered on all sided by the rice. Seal the container. Leave it for a week. (After that, seal the rice in 2 plastic bags and throw it out. The alcohol it absorbed makes it poisonous. Check with your local drug store about local laws regarding disposing of rubbing alcohol-soaked absorbents. Some places require you to contact your sanitation people, so it doesn't end up in the landfill, killing hundreds of birds.)
If you immediately remove the battery, you have a 50-50 chance of saving the phone, but I've saved phones that were in a lot worse than a few inches of melted ice, for a far longer time.
After a week in rice, the phone may turn on and seem to work, only to start failing over the coming weeks. That means that there was still some impurity that started causing a short and eventually something got fried. That usually means a new motherboard and other new parts (maybe a screen in your case). Since it's an S3, it probably will pay to fix it, but get an estimate first - either from a shop you know to be reliable by its reputation or from Samsung (they have people in Best Buy stores, in the cellphone area).
A technician (a good one, not a minimum wage clerk working in a cellphone store) will totally disassemble the phone, soak everything in alcohol, then "scrub" the parts with alcohol swabs (the kind they wipe your arm with when you get a shot) wrapped around toothpicks or other non-metallic "tools". Then bathe it again, then scrub it again ... until it's clean. Every part. Then dry everything in silica gel for a few days. Then reassemble it and test it. (And he'll be wearing at least 2 pairs of surgical gloves during the whole process - he has no idea where your phone has been, but toilets are some of the cleaner places I've rescued phones from.)
