Not with Verizon's email servers. Verizon's servers only support POP for communication between server and email client.s POP does not support synchronization of email across multiple devices.
IMAP allows for multi-device synchronization. You could subscribe to an email service that supports IMAP (most do) and forward your Verizon email to it, then you'd get what you want. Some popular free email services are gmail and outlook.com.