IMO, deer do not run towards water as a primary defense. Their first instinct when shot is to run towards heavy cover. Afterall, they do not know what has happened to them, they just know something is wrong and they are seeking security. A deer that is shot in the lungs or heart fortunately should never make it there, often dieing seconds from impact.
A gut shot deer is a different story, but has similiar tendancies. Just likea heart or lung shot deer, a gutshot deer knows somthing is wrong and will immediately seek cover. Since it normally takes hours to up to a full day in some cases for a gutshot to die, once the deer reaches cover and gains its composure, they then try to take care of themselves. Like it was mentioned before, if you got punched in the stomach, are you going to immediately run to grab a glass of water? No you are going to want to sit down and regroup. This is why when you track a gutshot, you will find a bed or beds where they have laid down to rest and regain strength. After a while of being gutshot, the bile and bacteria in the stomach and intestines of the deer will start to infect the area around the wound and the deer will gain a fever. To stay cool, the deer will seek out water to lay in. So in short, a gutshot deer will seek out water, but only if it doesnt die in one if its beds first.