Is it possible to delete my game center nickname?

Certainly! To change your nickname:

1. Open Game Center, and tap the "Me" tab on the bottom.

2. Scroll down and tap your Apple ID, then tap "View Account."

3. It might prompt you for your password; enter it.

4. Scroll down to the Nickname bar. Here you can change it to anything!

<Edited by Host>

Did you post it as a separate question? Let me know when you did and I'll be the first to answer!

Clarification: you can change it to anything that is not already in use.