Your immigrant ancestor who had not naturalized could not vote in elections. What else could your unnaturalized immigrant ancestor not do? The way to get the specific answer to this is to view contemporary state statute and see what rights are allocated to alien citizens. Unnaturalized aliens and those who had not filed a declaration of intent to become a citizen of the US could not file a claim for a homestead.
Depending upon the time period and the location, your alien ancestor may not have been able to file suit in court, bequeath property in a will, own property, etc. These laws have changed over time–do not assume what was true in 1800 was true in 1900. The answer is in contemporary state statute, many of which are on Google Books at http://books.google.com or Archive.org http://www.archive.org in addition to other locations.