User-caused damage of any sort is not covered under warranty, so you will have to pay. Apple does not repair iPods, but will replace the entire iPod for a flat fee; see the Repair Pricing section here:
There are independent iPod repair services that may be able to replace the screen for less. Do a web search for "ipod repair". Note, though, that if you elect to have the iPod serviced by an unauthorized shop, all further support from Apple will be void.