I assume you are asking about the gmail? Check your account settings. Sign in to gmail on your pc, click on settings(upper right corner), click forwarding and pop/imap. Under the pop section option #2, make sure it is not set to "delete gmails copy". I have mine set to "keep gmails copy" and can veiw mine on the phone until I choose to delete it.
As I have no experience with Verizon mail I can't give you any specific instructions. But I would still check the settings of the account. Most Pop3 push/pull settings have the option to delete upon read reciept.