According to what I can find in the code, the capability that is required to view this is edit_pages (an Editor roll capability) which I found in the class-all-admin-ui.php file but there doesn’t seem to be any filter or way to change it.
However, if your comfortable with code you can write a little plugin with this code in it:
function prefix_remove_pages_below_admin() {
if ( ! current_user_can( 'manage_options' ) ) {
remove_menu_page( 'activity_log_page' );
}
}
add_action( 'admin_menu', 'prefix_remove_pages_below_admin', 100 );
This checks to see if the current user has permissions to manage_options (an Administrator roll capability) and if they do not, it will hid the menu. However if the user knows the url they can still get there, and its not a built in solution, so its not perfect. But it will work until the functionality is added.
