Could an icicle with a broken tip be able to penetrate/pierce through a person's skin, or flesh just like that of shattered glass shards?

Probably not in the way you're thinking. 

It is possible to cut yourself on ice, particularly on shards that have sharp edges. I've seen people requiring a few stitches. If your hands aren't protected, then your skin can 'stick' to ice and peel it off if that ice is forcibly pulled away. Again, that's an injury I've seen in real life as a nurse in the ER. I've also seen people with burns from skin being in contact with ice too long.

I've seen reports of people being killed by falling ice, such as icicles dropping from buildings, but these deaths are due to heavy weights dropping from a height.

The problem is that human skin is actually pretty tough. To puncture through it, you need to apply a reasonably large force over a small area ... think needles and scalpels and knives. An 'ice weapon' also would need these pointy tips or thin edges to create puncture wounds. So, I'd imagine if you had an piece of ice with such structures, they'd be too thin and brittle to do any real damage ... in other words, these bits of the ice weapon would snap off.

If your weapon was heavy enough and you applied enough force then I'd imagine the kinds of injuries would be more like bruising or broken bones ... more 'battering ram' than 'glass bottle'.