This is such an annoying error because it seems as though it should be such a simple thing to get working.
Well there is a simple solution, which I will get to later (and I don't mean enable anonymous access!). In the meantime you are going to be wondering what the problem is, because it's not as though you are trying to go to an external site or anything (which probably works). Surely getting a feed from your own site should be easier, not more difficult.
Of course the general statement "The RSS webpart does not support authenticated feeds" is not strictly true. The RSS webpart does not support some authenticated feeds. And of course the RSS webpart does not support your authenticated feed, which is the most important thing. It's a variant of the old "double-hop" authentication issue. That means NTLM is not going to cut it - you'll have to find another way (see below). On the other hand if you are using Kerberos authentication you should be able to get it working, so check the following:
So I promised a simple solution, and here it is: don't use the out-of-the-box RSS web part. This is such a simple thing to write if you have any development skills at all. I have my own version that I keep using on projects. And if you don't want to write your own, there are hundreds of code samples and complete solutions you can download for nothing on the web. Or if you want a bit of commercial back-up, there are quite a few third-party commercial offerings too. As far as I know they all work perfectly.