I've been trying to get past it for possibly 20 minutes. I can't hit it with a skyward strike or my slingshot. Fi says that it looks at pointy objects, but the only things that I can get it to look at are me and my sword. What am I supposed to do?
You're supposed to make it dizzy by rotating your sword in a circle (not a spin attack). Do it quickly, but not too quickly—if you go too fast, you'll start doing random swings, which won't work.
For the eyes that appear later in the temple, you need to have all of the eyes in a given room look at you at the same time in order to get past them. If not all of the eyes are open, they won't get dizzy and die no matter how fast you rotate your sword.
I'll start off with a hint, with the the solution posted in spoiler tags directly below it.
HINT : Make it dizzy.
You need to move your Sword around in a circular pattern until the eye turns red and falls off, in other words you're making it "Dizzy". If there is more than one eye, you need to do this to all of them at the same time, you cannot take them out one-by-one. Therefore if one of the eyes are closed, then no matter how long you stand there any eyes following your sword won't become dizzy and fall off. The best way to do this is to stand in the centre of all the eyes.
Here's a video showing the solution:
Unlike in past games, you don't have to hit the eye. You have to sort of hypnotize it. As Fi says, eyes look at pointy objects, so they will stare at the point of your sword. Get your sword out and make a continuous circular motion for a couple of seconds while the eye is looking at it, until it "dies".