The halting problem requires a deterministic system, it does not require that the universe itself to be deterministic. The universe encompasses everything that ever was, is, or will be... including all deterministic systems.
I'll accept that definition of the universe, but recognizing that "all deterministic systems" might be an empty set.
My main point is that in a deterministic universe you should be able to contrive a deterministic thought experiment which will always be able to correctly predict the outcome of the experiment, but if you design the experiment so that its output is always the opposite of whatever was predicted, then it becomes evident that there can never be sufficient information at the beginning of the experiment to predict its conclusion, and if the current state of the universe is not sufficient to predict a future state, then the universe is not deterministic.
You're basically trying a proof by contradiction here, I believe. However, your wording is really loose.
First, what do you mean by "predict the outcome of the experiment?" What experiment?
Then you say that you "should be able to contrive" an "experiment" that can predict the outcome of the "experiment," but that it will have the "output" that is the opposite of whatever was predicted. What does this statement even mean, since it appears self-contradictory, like saying let the set A contain the number 5 and not contain the number 5. I can't tell whether your conclusions follow from your premise as your premise seems confusing at best.
It seems like you're trying to do something like this: Let f(x) = x + 3. So, f(5) = 8. So let's redefine f(x) = x + 3 if x5, and it equals 9 if x=5. You haven't made a contradiction, you've just swapped one deterministic function out with another.
You also brought up the halting problem, and it isn't obvious to me how that relates. The halting problem has nothing to do with determinism, just predictability. The behavior of a halting machine is completely deterministic - it has a finitely definable starting state, and given that starting state will always end up executing the same series of steps. However, it is unpredictable in the sense that you can't tell what the result will be with certainty without actually running through all the steps to get there.