Welcome to Android Central! Watches like that one are not Android Wear, so they won't necessarily do the things that Android Wear watches like the Moto 360 and ASUS Zenwatch are known for. Your watch is basically a miniaturized phone on your wrist, and adding a SIM card would make it a standalone phone that you can make calls with. The problem is that it's unlikely to have a CDMA radio (which is what's needed for you to make voice calls on Verizon), and besides, Verizon is picky about which devices it allows on its network. You'd probably have to start a new plan with a GSM carrier (like AT&T or T-Mobile), but even then, it depends on what network bands the watch supports, since it has to be compatible with the ones the carrier uses. Look in the documentation to see if it's mentioned, or search the web for that info.
It's possible that pairing the watch to your phone via Bluetooth may let you speak to and hear other callers using the watch, as long as your phone is with you. You'd have to look at the manual to see if that's possible.