It is possible for Apple to replace your screen, although it will cost a lot. When I dropped my phone on a sidewalk and badly cracked the screen, I went to a place that fixes phone screens but isn't associated with Apple, and it was only $60 as opposed to the $129 that Apple would've charged me. However, there are disadvantages to using a third-party phone repair place. Any warranty on your phone will be voided, and Apple won't touch it with a ten foot pole. If you can trust yourself to keep your phone safe and sound, and don't care about the warranty, I would recommend using a third-party retailer, but if you're willing to spend more money and care about the warranty, then just go ask Apple to help you.
I would also recommend purchasing a protective case such as Lifeproof or Otterbox. Personally, I trust Otterbox more (have used both types). Good luck with your phone!