Hi Michael,
Thanks for posting in Apple Support Communities about this. It sounds like the Gmail server is having issues being trusted, alerting you of the issue repeatedly. We'll do our best to assist.
In the second screenshot, it states the certificate isn't 'trusted'. Are you able to select on More Details to give it permission? If not, then this looks like it's happening for the Calendar in Gmail. Go to Settings > Calendar > Accounts, select on your Gmail account, then disable and re-enable Calendar. This should push any specific settings from Google to the device and straighten it out.
If that doesn't work, then try signing out of Google and back in to your account. You can do this in Settings > Calendar > Accounts, select on your Gmail account, then Delete Account. Go back to Accounts, then Add Account to sign back in.
Should it persist, you may need to contact Google in order to make sure the account is in good standing and if there are any outages in your area: Google Help
Wishing you the best, take care!