Can I install a WordPress site locally from a tar.gz file?

Can I install a WordPress site locally from a .tar.gz file?

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.