Spent hours finding this, and for some reason most of my forum searches for things like new_products.php and TABLE_HEADING_NEW_PRODUCTS were unhelpful due to lack of replies or outdated responses, so to save someone else a little time...

If you'd like to stop the New Products box from displaying on the main page of the store, you can edit
Code:
catalog/templates/fallback/content/index_default.tpl.php
and comment out the line
Code:
<td><br><?php include(DIR_WS_MODULES . FILENAME_NEW_PRODUCTS); ?></td>
like so:
Code:
<!-- <td><br><\\?php include(DIR_WS_MODULES . FILENAME_NEW_PRODUCTS); ?></td> -->
Note two comments - one for the HTML and one inside to kill the php.

This is on a store using aabox template and by default the fallback content/index_default.php template because I've added none of my own.