I will interpret the question as suggested by Philip: whether an argument to this effect has been discussed in philosophy. It was. A philosophizing mathematician/computer scientist/physicist David Wolpert formalized just such an argument in Physical limits of inference. Wolpert formalizes measurement, observation of a phenomenon, memory of past information and prediction of a future state in terms of inference devices that are independent of the laws of physics. A yes/no question must be posed to such a device, and, based on whatever apparatus it has at its disposal, it provides an answer. There are no limits on the computational power of the device, but it must be "within" the universe it makes the predictions about, in a precise sense. As the OP suggests, the proof is a version of the Godel's incompleteness argument.
A popular exposition, without mathematical technicalities, is given by Scientific American under a catchy title Within Any Possible Universe, No Intellect Can Ever Know It All:
"Wolpert proves that in any such system of universes, quantities exist that cannot be ascertained by any inference device inside the system. Thus, the “demon” hypothesized by Pierre-Simon Laplace in the early 1800s (give the demon the exact positions and velocities of every particle in the universe, and it will compute the future state of the universe) is stymied if the demon must be a part of the universe...
The theorem’s proof, similar to the results of Gödel’s incompleteness theorem and Turing’s halting problem, relies on a variant of the liar’s paradox—ask Laplace’s demon to predict the following yes/no fact about the future state of the universe: “Will the universe not be one in which your answer to this question is yes?” For the demon, seeking a true yes/no answer is like trying to determine the truth of “This statement is false.” Knowing the exact current state of the entire universe, knowing all the laws governing the universe and having unlimited computing power is no help to the demon in saying truthfully what its answer will be.
The proof is mathematically impeccable, but there is a catch, already discussed in austinlorenz's answer to Have David Wolpert's findings really “slammed the door” on scientific determinism? The "within" stipulation turns out to be a way of allowing self-reference, and the "question" is no more meaningful than the Liar sentence with the question mark at the end. There is a state of the universe that an inference device cannot predict, yes, because the answers it is designed to give can not also be their negations.
There is an alternative interpretation of the OP question suggested by the remark "whatever the computer printed out, the person could do something different". This is more or less the key ingredient of libertarian free will: the freedom to do otherwise in identical circumstances. Whatever else it means or entails, it certainly entails indeterminism. And if we stipulate indeterminism we can certainly infer it: no accurate prediction is possible by stipulation. But the contrapositive is equally valid: if we do not assume indeterminism then determinism is not ruled out, and accurate prediction is possible.
This is similar to Novikov's self-consistency principle for time travelers: one can go to the past as long as the present already incorporates what they "caused" there. In a popular illustration, a time traveler, who went to the past to kill her own grandfather, slips on a banana peel strategically "placed" by the deterministic universe at the critical moment, and fails.
This is why the inside/outside distinction (we can only do otherwise if we "know" the prediction) either does not work, or introduces the same sort of circularity as in Wolpert's argument. It is not that the prediction has to stay idle, and not "cause" anything "inside", in which case it will be "outside". It can "cause" things around at leisure, the point is that all of its "causing" is incorporated into what it predicted anyway. There is no need to "know" it, or refrain from acting on such knowledge, the only question is whether the universe is deterministic or not. If it is indeterministic the impossibility conclusion is baked into the premise. And in a deterministic universe things will happen the predicted way regardless: trying to make it not happen will cause it to happen.