Calendar invite to someone using gmail address for iCloud account

That's odd. I just sent a sharing invitation to a Gmail address I have (which is not an Apple ID address) and it arrived in my In box, and very quickly too. Bear in mind that this address is in no way connected to an Apple ID.

As the invitations come from Apple's iCloud calendar service - and not your email client - I'd have to conclude that the problem is with your friend's email service. Could the email ended up in a spam or junk folder? Has your recipient looked at their Gmail on the web, rather than trust it to their device?

Can you expand on what you are trying to achieve? Is the recipient already sharing the calendar? If you are sharing an iCloud calendar with somebody all the events in that calendar will synchronise with their calendar also, and would not require an invitation to an event.

You're very welcome.