When I try to download Bitcoin Core on MacOS it says "Apple cannot check it for malicious software"? Is it safe to download? Brand new Mac Mini, latest macOS Catalina (10.15.6 (19G2021)), followed instructions on bitcoin.org download page.  Double click Bitcoin Core.app, and I get:

“Bitcoin Core.app” can’t be opened because Apple cannot check it for malicious software.
This software needs to be updated. Contact the developer for more information.

Any suggestions?
As mentioned by Andrew, this is due to macOS' enforcement of notarization requirements.
This can be bypassed by right clicking on the application, then selecting Open - A dialog will popup warning you that it is not verified, but this one will have an Open button.
Once you open the application through the above process once, a regular double click will work on it in the future.
You may need to go through this process again if you update to a newer version of Bitcoin Core.