Is there a way to block a specific person from leaving voicemail?
Outside of having Verizon block their number entirely no. The VM is handled within Verizon's system so I see no way to control this from the phone.
But I'm on AT&T.
My bad. I think the same thing applies to AT&T though.
Look for mr. number app. You will be amazed. Can send directly to VM or simply answer and hang up the call automatically not giving the chance to leave a voicemail. Hope this helps you.
Google Voice can do this.
You can block a caller and they get a generic message and can't leave a message when they call your carrier mobile number. Your phone will still ring, but they are not able to leave a voice mail message. You can also set a contact in GV to just go straight to VM, so if you do that and block them I suppose your phone won't ring. I will have to try that...
If you block someone and they call your GV number your phone will not ring at all, nor can they leave a message.
Separate names with a comma.