This answer may be specific for SharePoint 2010:
Using javascript (even better with jQuery) you can accomplish this.
On the webpart page, your page viewer part will contain a div element with a unique id (e.g. WebPartWPQ3). Inside this div is an iframe that points to the page to display (i.e. src="your page"). You can use your favorite DOM inspector to find this div and get the id.
Using javascript/jQuery, you could reference the div with that unique id for your web part, grab the iframe, and then change the src attribute to point to your new page.
Untested snippet:
<script type="text/javascript" src="http://sp2010/_layouts/smit/jquery-1.8.3.min.js"></script>
<script type="text/javascript">
function ChangeUrl(newurl)
{
var pg = $("#WebPartWPQ3 iframe");
pg.attr('src',newurl);
}
Have another bit of javascript/jQuery that responds to your URL textbox change event that calls the above function (passing in the value for the new URL from the textbox as the newurl parameter).
Forgive me for not testing the above snippet (my SharePoint box is out of commision at the moment). But the jist of the above should work.