Hi @Bman9111 ,
Is there no way to remove the public calendar option within Exchange owa?
As far as I know, this is currently not feasible. As mentioned in this official blog, once the calendar is published for anonymous users, it allows "anyone with the link to the calendar to view it without having to log-on". (The blog only mentions about Exchange Online and Exchange 2013, but can be applied to Exchange 2019 as well.)
And for your scenario, seems like setting up federation is not applicable, right? Given this, I am afraid there's no way around it at present.
I've tried submitting this idea to the official feedback portal for Exchange server. The link would be left below in case you or other community users would like to add vote or comment there as well:
https://feedbackportal.microsoft.com/feedback/idea/81a352b0-4eb6-ed11-a81b-000d3a0450e3
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