Yep.
I apologize for not making it clear, but there are two scenarios floating around.
The first is of a "turing equivalent" machine, like a computer, being set up basically like we are now -- you have some interface with the external world and the machine walks around and does stuff and the only thing that is "computed" in a strict sense is whatever is inside it, kind of like the only things that are "thought" are inside your head.
Now even in this case, it is impossible to have full self reference -- you are not aware of the individual neurons in your brain. In fact you are not even aware of your brain at all. All you know is that you have a head and you seem to see and hear from it.
Also in that case, however, you don't need everything to be calculated. Obviously, the real world provides the vast majority of information needed, just like your brain doesn't "think" about all the stuff reaching your senses before it gets there. A tree falls independently of you -- you just observe it.
But this isn't a strict turing machine. It is just "turing equivalent," which means it can't do anything a strict turing machine couldn't do given a tape of infinite length -- which is impossible. So it just means "it can pretty much be rigged up to follow any set of instructions and make any set of computations you can think of."
It isn't a "strict" turing machine because the ideal notion of a turing machine includes only the machine and a very long tape that represents both input, output, and memory. In other words, you can't continually feed it input in real time because the tape is set at the beginning of the run -- everything that would be input has to already be on the tape somewhere. Thus the tape is a little self contained world of information -- there is no way in or out once the run starts. That is why even a very simple "turing equivalent" machine in the real world -- like a small collection of neurons wired up a certain way -- would require an infinite amount of tape to account for all the stuff they could encounter in the real world. Everything "else" in the real world needs to be included in that initial tape. Clearly not realistic, but it is supposed to be abstract anyway.
In that case -- a strict turing machine -- the machine cannot reference itself. In fact the machine cannot reference anything. What references stuff is whatever is on the tape. It only makes sense to say some information on the tape references information somewhere else on the tape. The machine itself is not included in the information any of the calculations have access to. If you want to talk about Godel and incompleteness, it is "outside" the system. Anyone in the tape has no access to it and could not explain its existence any more than we could explain what the first cause of the universe was.
In such a scenario, if the machine were running the tape and making the computations that represent a brain in some world, it is not the machine itself that is conscious but rather the information on the tape. This is analagous to you being conscious instead of the universe being conscious even though the universe is the thing "making the calculations" to move particles around in your brain.
The way these two scenarios -- the robot vs. the ideal turing machine -- are linked is that *if* a robot is conscious, then because all of its input must be converted from the real world into digital signals -- things that can be computed -- in principle all of that input could just originate as a computation to begin with. In other words, you could take the computations being performed in the robot brain, move them to the tape of an ideal turing machine, then move all of the input data as well to the tape of the turing machine, and start a run. How would the robot know the difference? It wouldn't. Kind of like how we can't tell if we are in a simulation or not.