The Updated Pages side navigation on the Wiki Pages Library is not related to the current navigation. To remove this you must either override the CSS via a Script Editor Web Part, or modify the underlying Master Page.
I'm re-posting the correct answer from http://stephen-white.blogspot.com/2015/05/hide-recent-updated-pages-in-sharepoint.html:
If you have implemented a SharePoint 2013 wiki page library for your
company, you might agree with me that the recent and updated pages
links are an unnecessary distraction to your users. Just because a
page has been updated, that doesn't suddenly make it more important
than any other content in the site. The problem is that Microsoft
doesn't give us a supported way to remove these links.
So, we add a small piece of code. I like to have big bold commenting
on anything I add so that I can easily spot it later on.
<!-- REMOVE THE LEFT HAND NAVIGATION PANEL -->
< style type="text/css">
#sideNavBox { display:none; }
</style>
You can either add it to a Script Editor web part on each page, or
install SharePoint Designer 2013 on your PC and add it to the master
page.