Since it's not jailbroken, it's not malware of any kind. There's no known malware capable of infecting an iPad that hasn't been jailbroken.
In addition, it is a common internet myth that such behavior is the result of a virus. In reality, no hacker wants to go waving the cursor around under the user's nose. (Or, in the case of an iOS device, tapping and navigating, since there is no cursor.) This is almost never the result of malware on a computer, and definitely never in the case of an iPad.
First, try resetting your device. Press and hold both the home button and the sleep/wake button for about 10 seconds, until the screen turns off. Ignore the slider that will appear. Release the buttons, then press and hold the sleep/wake button until the Apple logo appears. This is the equivalent of a reboot, and should not affect your data or apps.
If that doesn't help, try restoring your device from backups, and if that doesn't help, restore it to factory settings (ie, do not restore a backup). See:
http://support.apple.com/kb/ht1414
If none of that helps, the issue is caused by some kind of hardware problem, and the device will need to be repaired or (more likely) replaced.