Might depend on the phone.
Most phones come with software (either on CD or downloadable and sometimes you even have to pay for it) that allows you to transfer data from the phone to a PC (by USB-cable or IR wireless or Bluetooth wireless). That data might be ringtones or pictures or movies or mp3's or the address book or the text of SMS's or recorded phone calls or recorded messages like this one.
Nothing can be said about the possibilities without knowing exactly what phone it is.
And there's always the possibility that the voice mail message doesn't reside on the phone itself, but somewhere on the Verizon-network. That's certainly the case if it's a message of the "Hi, this is me, my phone is off at moment, please speak your message after the beep" type, because a phone that's off won't do anything and certainly won't deliver a message.
If the message isn't on the phone itself, there's no way to get it from there. All you can do then is to contact Verizon customer support and ask if they can send you the sound file from their network, before cancelling the subscription (and thus deleting it).
Hope this helps.
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