An icicle of any sharpness would be more than sufficient to gouge out the eyes. With sufficient force and a large enough icicle (for strength), one could possibly fracture the skull and destroy the brain (sharpness would not matter once it enters the eye socket, and the icicle would turn into a stake at that point). However, icicles that can be wielded as weapons lack the characteristics necessary to pierce skin - handheld icicles are too weak, with dull points lacking the piercing power and sharp points breaking too quickly. Attempting to stab someone with an icicle would amount to stabbing someone with a new, unsharpened (aka flat end) pencil. With sufficient mass and force, an icicle *could* pierce skin, but the forces required would be impractical for handheld use.
Icicles are extremely brittle when used as blunt weapons. When swung like a club, the end that makes contact with a target will almost always snap away from the "handle." However, thick icicles can withstand the force of being swung hard - so, icicles pose a powerful one-strike weapon. An icicle wielded with two hands (two to three feet long and of sufficient width) could easily break bones or kill with a blow to the head. Handheld icicles (one to two feet) would have a similar effect to using a glass bottle as a weapon, capable of incapacitating or heavily bruising. Even when broken, chunks of the icicle can be used like rocks.
As a stabbing weapon, icicles are only practical on easily subdued or already restrained victims. Icicles of practical size are incapable of stabbing through most skin and clothes. The eyes are the only easily stabbed points, and striking those in active combat would be a notable feat. As a blunt weapon, icicles are more useful, capable of killing or severely impairing with a first swing and retaining use as blunt weapons after shattering.
All in all, an icicle would be a weapon of eccentricity or improvisation.
In my knowledge, I have heard of two people killed by falling icicles (no murder). Both were killed as they were removing icicles from buildings and unwittingly causing a chain reaction of falling ice - the only difference is that one was a college student fooling around, another was a homeowner trying to prevent a roof collapse after a heavy snow/sleet.
You can refuse to think, but you can't refuse the consequences of not thinking.