Any external 301 or 302 redirect to the other page is going to be shown as the current location in the browser URL address bar. To achieve the result you want you have to keep the current parent document as your local index.html. So you have a couple of options to do that. (probably not a complete list...)
wrap the request in a ProxyPass
(this is probably the most transparent to the end user, as they have no way to detect that the page is actually remotely hosted, but is slightly more ball-achey)
To hide the redirect from the client, you would need to Proxy the request in the httpd.conf
file for your VirtualHost like so;
<Location /index.html>
ProxyPass http://www.someothersite.com/blah/blah2/blah3/index.php?user=80338
</Location>
To enable the above directives, you are going to need to install and enable the mod_proxy_http
apache2 module, which is system dependent (e.g. yum, apt, a2enable)
However, you are going to get some interesting URL mapping issues, which you would have to tackle on a case-by-case basis, so to get this to work transparently would take some effort.
For example it depends how the links in your remote document are specified relative, or fully-qualified. You can fix each link and URL using ProxypassReverse and Mod_substitute rules.
Alternative: Load the page into an iframe (or use old skool frame)
You can use an iframe, which would be shown as the browser URL address bar, and load the remote document into an iframe with something like this;
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="EN">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>Full Page IFrame</title>
<style type="text/css">
html {overflow: auto;}
html, body, div, iframe {margin: 0px; padding: 0px; height: 100%; border: none;}
iframe {display: block; width: 100%; border: none; overflow-y: auto; overflow-x: hidden;}
</style>
</head>
<body>
<iframe id="tree" name="tree" src="http://www.someothersite.com/blah/blah2/blah3/index.php?user=80338"
frameborder="0" marginheight="0" marginwidth="0"
width="100%" height="100%" scrolling="auto"></iframe>
</body>
</html>
use old fashion frameset
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Frameset//EN" "http://www.w3.org/TR/html4/frameset.dtd">
<html>
<head><title>My First Frame Page</title>
</head>
<frameset cols="100%">
<frame src="http://www.page3.com">
</frameset>
</html>