Does network reset delete eSIM and require Verizon?

Resetting Network Setting on an iPhone will erase all saved Wi-Fi networks and passwords, cellular network settings, VPN configurations, and APN settings. However, it does not remove or disable eSIM profiles that are already installed on the device. If you want to delete an eSIM, you must remove the eSIM profile separately from Settings > Cellular (or Mobile Service).

Summary:

• Saved Wi-Fi passwords are removed from the device (unless they are restored later through iCloud Keychain synchronization after you reconnect).
• Bluetooth pairings are also removed, so you'll need to pair Bluetooth accessories again.
• Installed eSIMs remain on the iPhone and can still be used after the network settings reset. You may need to reconnect to Wi-Fi or allow the carrier settings to refresh, but the eSIM itself is not deleted.
• The only way to remove an eSIM is to choose Delete eSIM (or Remove Cellular Plan) in the cellular settings, or to erase the iPhone and explicitly choose to remove eSIMs during the erase process.

Depends on the choice you make.

When you reset all settings, you can choose not to delete the eSim, also, if you do delete it, it should be possible to just install the esim from the Verizon App. Your service plan or line number will not be deleted from Verizon.

https://www.verizon.com/support/knowledge-base-308606/