An app by itself isn't going to provide the solution you're looking for. Voicemail isn't a phone feature. If you're using the standard voicemail then it's Verizon's voicemail system. Unless they designed it with API's or some other method for apps to access voicemail (highly unlikely) then an app-only solution is out. You'd need a repalcement voicemail system, like GV (or any of the others).
Unless, maybe, the app dialed your voicemail and then checked and transcribed your messages. It would be a terribly kludgy solution though.
Instead of the cell phone provider being the one who could track or listen on your calls or whatever, now it would be Google who would be able to track everything you say, every conversation, and every text.
On your Google phone. With a GPS that knows where you are. And they have your credit card if you buy apps.
