No - the method by which your Mac connects to the internet doesn't affect whether it can hand off calls to your cellular service in general. I've tested this mostly in the US with US carriers, but it's possible you have different rules and would need to check with your carrier.
I suppose your network could be broken or firewall or traffic is preventing a properly set up set of devices from working, but the more common cause is the carrier setup didn't finish.
I would focus on undoing and re-doing the steps needed for the carrier to talk to the iPhone and then the settings on the iPhone to enable WiFi calling.