Is rigged Hilbert space generally considered the correct structure for QM? I am currently reading the quantum mechanics text by Ballentine and, over and over, arguments are made (e.g. Chapter 4.6 on constraining the wavevectors of free particles to be real) which rely on rigged Hilbert space as the "correct" context for a quantum theory. Is this generally accepted as true? I ask this because I have this vague notion that there exists another capitulation/mathematization of quantum theory which used operator theory/spectral theory/follows von Neumann, and it seems to me that this is a completely different approach than the rigged Hilbert space approach. Thus my question, overall, is whether there is consensus from mathematical physicists as to which is "most correct", or are they perhaps equivalent?
 A: I do not think there is a unique answer. It mostly depends on personal taste.
However, I think that almost all mathematical physicists agree on the fact that the QM approach based on the rigged Hilbert space structure requires more mathematical hypotheses, and it is much more delicate to rigorously handle, than the von Neumann formulation. Which, on the other hand, can be immediately applied to more general quantum theories than QM, whereas this extension is not similarly easy for the rigged Hilbert space formulation.
Personally speaking, for that reason I strongly prefer the von Neumann framework (though I am conscious that it may be dangerous if used uncritically).
In all my career, I never met a colleague who really used  the rigged Hilbert space for rigorous computations.
However a distinction is necessary between Dirac's improper eigenvectors formalism in QM and the  theory of rigged Hilbert spaces to formalize QM.
What is evident is the formal powerfulness of the non-rigorous  Dirac formalism. That is much more powerful, for practical manipulations, than the rigorous and solid approach by von Neumann.
The rigged Hilbert space formulation, essentially due to Gelfand, is a fruitful attempt to demonstrate that Dirac's manipulations can be formulated into a rigorous setting.
However physicists are not interested in that rigorous re-formulation of something already evident, on the one hand.
On the other hand,  mathematical physicists do not need it, because von Neumann formalism is already pretty enough.
For that reason the rigorous rigged Hilbert space formulation is a sort of strange animal to admire in the zoo of mathematics applied to physics. Its existence is a proof of the fact that when the physical ideas are really good, then they can be formulated in a rigorous mathematical framework (the converse fact is tragically false).
The safe view on the issue, in my honest opinion, is to use Dirac’s formalism to guess the physical result and, if one also  needs a rigorous proof of that, to pass to the von Neumann formalism to consolidate the guessed result.
All that, as a byproduct, leads to an illustration of the difference between mathematical physics and theoretical physics. The former proceeds in terms of theorems, so that the mathematical rigour is strictly necessary; the latter mainly looks at the physical plausibility of achieved theoretical results, without paying much attention to mathematical rigour. Both use mathematics, but for the former it is a guide, for the latter it is a slave.
