Can I install a WordPress site locally from a
This likely depends entirely on the contents of the archive.
.tar.gz is just a compression format. You will have to extract it to have any website work correctly but otherwise the particular storage format has no bearing on your question.
I created a backup, which produced a
.tar.gz file. I also downloaded the public folder as a zip (has wp-content, wp-includes, etc).
Wordpress typically requires a database to run. You don't mention whether or not you have a backup of this, but if you don't, you may run into issues.
I am redesigning the site, but I need to be able to view these old pages/content.
In theory it's possible to reproduce a site locally. But dynamic sites like those produced with Wordpress may be more difficult.