And I'll say yes 🙂, you could, but.. (Yes, there is a but).. It depends a lot on what your website actually consists of.. And as we do not know your code, it is a bit dificult to judge which trouble you run into..
Assuming that the index.php holds the formatting of the page, and that includes the menu pages, You'd have to add the formatting code to each page you want to show. You'd then have a static website, which does not use the full potential of PhP, and PhP is basically only used to avoid having to update the menu every time for every page.
IF you are running it from a database/using file-content includes, then you would have to pass parameters to actually know which file to get / record to retrieve from the database (And that is what the ?page_id=28 does). But you could use mod_rewrite to use:
..index/page/28
instead of
..index.php?page_id=28
However, this takes some study.