Avast is blacklisting ESO’s launcher from updating and I would like to know how to stop it from doing that.
It’s not so much blocking the launcher but blocking the site URL:Blacklist.
Your image appears to be from the notifications area, did you get an Avast alert window ?
Reporting Possible False Positive File or Website - https://www.avast.com/false-positive-file-form.php.
As a temporary fix, try entering the url in the exclusions. under settings > General> Exclusions. This should allow the updater to function until corrected by the virus lab team