Can't access Mail or Mail setttings on iphone 4 + iOS 6.1.2

Solution 1:

First, force-quit Mail.app. To do this: Go to your home screen. Double-click the home screen. Locate Mail.app in the bottom bar that appears. Press and hold Mail.app until the icons wiggle. Delete Mail.app from the bottom bar by clicking red X.

Try again to launch Mail. If this doesn't work, reset the iPhone by pressing and holding power and home buttons for 10 seconds. Restart device once it shuts off.

Solution 2:

• you could try this, though it's unlikely since you can see the mail icon

  1. Try going to the Settings.app.
  2. Once inside: General>>
  3. scroll down and select "Restrictions" (this is at the bottom
  4. Make sure Mail is both 'allowed' ('On') and that Restrictions are disabled, unless you want them enabled. (Restrictions are disabled if all your apps are light in color/unsaturated (grayed out), and the button at the top of the screen reads "Enable Restrictions").

• second, you could do a factory reset/restore, then restore from your latest or last known good backup. Hopefully your mail messages exist in such a way that you can reset/restore without losing anything. (Depends on mail settings. If you are using POP and you have unsent drafts you will likely lose them. If you are using IMAP you should be fine, but I can't guarantee this.

• I would definitely do the steps bispymusic suggested.

• to clarify,
  1. Using the "Slide to power off" then holding both buttons to turn the device back on is called a 'Restart', and is considered more topical.
  2. Holding both buttons while the device is on until the logo shows up again is called a 'Reset' and is used for more severe ('deeper') issues that don't go away after a 'Restart'.