When you tap the Mail icon .. does the Mail app open at all or does it go to a white screen for a few seconds then shut down ?
Have you tried a reboot ?
Holding down the Home/Sleep buttons at the same time until the Apple Logo comes on.
If you have tried those .. try going into Settings - Mail, Contacts, and Calendar and turn off Mail for all accounts then reboot. <--- This has worked for some.
My iPhone Mail app won't open, but all the settings are still in there and correct. It's a iOS 6.1 issue.
Right now my only option is another Mail app it looks like.
Hope this helped.