For sure you do....but Verizon may be in a tight spot here. Just last week the Federal Govt revised the method by which my former employer (NSA) retains access to your emails.
For your protection, all correspondence (written and spoken) is archived somewhere. The new place is in limbo - so I suspect all phone companies are covering their behinds until the new plan has been clarified to them. In the immediate past, a massive server farm in Maryland held all that data. It is in a new state of transition and I suspect that your desire is just not possible at this time. I suspect that it can be hidden but not deleted.