Is it possible to open a HTML page inside UNITY, with the page functioning normally?
It depends what you mean by "the page functioning normally". Assuming you mean a browseable webpage within a game, then no.
You can make HTTP requests from Unity and thus download the data, but Unity isn't a web browser and won't display the HTML as a webpage. Similarly, Unity can open a web browser to a specific URL, but then that's not "inside Unity".
Use the WWW.Load () feature.
You'll need to know the file path on the users device.
It's trivial to load information and files from a url online into unity by passing json, xml, material types (unity doesnt have a browser itself). Alteratively, you can pass the user to their browser to view the webpage.