Why does LastUpdatedOn get updated when I simply view a document?

There are a couple of things that can potentially be triggered on document view that will cause LastUpdatedOn to be set. First you may be setting the 'last action date' on view.  

Also, in your Record Type, you may have chosen to log 'Document Viewed'.

To avoid setting LastUpdatedOn disable both of the above and then access the document like this:

Record rec = new Record(database, "REC_1");

if (!rec.IsDocumentInClientCache)
{
    rec.LoadDocumentIntoClientCache();
}

Console.WriteLine(rec.DocumentPathInClientCache);

BTW...

Yes, I agree, it does seem odd to update LastUpdatedOn simply because we log 'Document Viewed'.  I have put in a request to see if it is possible to change this behaviour in a future release.