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It is well known that the observed energy density of the vacuum is many orders of magnitude less than the value calculated by quantum field theory. Published values range between 60 and 120 orders of magnitude, depending on which assumptions are made in the calculations. Why is this not universally acknowledged as strong empirical falsification of quantum mechanics?

Experimentally, based on cosmological observations, there seems to be a vacuum energy (the "dark energy" component of the cosmological energy budget), with a certain value. A the present epoch, there seems to be about three times as much vacuum/dark energy as there is "dark matter" and about fifteen or twenty times as much dark energy as there is visible matter. This concentration of dark energy poses two very serious puzzles, but neither of them is at all suggestive of a breakdown of quantum mechanics.

The first problem, mentioned in the question, is the "hierarchy" problem. There is no quantum mechanical prediction for the absolute energy density of vacuum. However, it is possible to make some very crude "guesstimates" about this quantity. We know that some new fundamental physics must take over at the Planck energy scale $E_P$, where gravitational interactions are in the deeply quantum regime. We may therefore guess that the vacuum energy density is proportional to $E_P^4$. (This is certainly not a prediction of quantum mechanics though. Strictly, according to quantum field theory, without including gravity, the energy of the vacuum is unobservable and therefore not even well defined.) The problem with the $propto E_P^4$ guess for the vacuum energy is that it is off by 275 nepers or so. But that does not falsify quantum mechanics, since our guess was not based on rigorous quantum theory anyway.

The other puzzle with the vacuum energy density is that its value at the present cosmological epoch is pretty close to the energy density of matter in the universe, even though there is no a priori reason why the two should be related. The fact that the (light plus dark) matter and dark energy densities are relatively close suggests that what we are observing as apparent vacuum energy might very well be something else entirely anyway. But what that "something else" might be, no one knows.

"Quantum mechanics" is actually a very general, broad theory that "really", at least working from many more modern understandings of the topic, is about information, and more specifically, it is a language for writing theories that describe (in some way) physics in which information content is limited, just as relativity is actually a theory of space and time in which information propagation speed is limited. And moreover, that the information is limited in such a way that there are trade-offs between information determining various physical parameters of a system, and not a simplistic "pixelization". That is, in effect, what the "true" meaning of Planck's constant $hbar$, and the fact that $hbar > 0$, means. Check out Scott Aaronson's page here for the idea of quantum mechanics as a language for writing theories, instead of per se a theory in its own right:

https://www.scottaaronson.com/democritus/lec9.html

though it doesn't specifically touch on the "information limit" notion, for that, try:

https://iopscience.iop.org/article/10.1088/0143-0807/36/1/015010

e.g. section 3.8, mentions the idea of QM as an information-limited theory, at least in touching, though doesn't quite go about it in the same way as I had worked it out.

The way then to "falsify" quantum mechanics would be to show an instance where its informational limits are violated, e.g. if someone finds a way to create a particle that has position and momentum (or another pair of incompatible physical parameters) more precisely defined than Heisenberg's limit allows. Merely finding a failure of certain theories built on it (e.g. "quantum field theories" - QFTs) to account for a cosmological parameter's value which is already going to be well in the range of those limits is not going to necessarily falsify QM, as another theory written in its language might still work and be able to account for that result, even spectacularly. It will simply falsify that particular theory built using it, namely Standard Model QFTs. (Whether QFTs entirely are out, at at least a fundamental level, is disputable, but the SM is at least guaranteed to have something wrong with it.)

Our back of the envelope prediction for the order of magnitude of the vacuum energy is indeed very wrong! However, keep in mind that

1. It is possible to precisely fine-tune free-parameters of the theory to match the measurement. This is achieved through a delicate cancellation between so-called tree-level parameters and corrections. When we make the back of the envelope calculation, we implicitly assume that such cancellations don't occur.




2. This isn't a test of quantum mechanics per se; but a test of a particular theory that obeys a combination of quantum mechanics and special relatively. Such theories are called quantum field theories. There are many such theories as we may introduce lots of types of fields and let them interact in lots of different ways.

So, quantum mechanics isn't falsified as measurements of the vacuum energy don't directly test it. And even the theories that the measurements do test aren't falsified because we can find extremely fine-tuned combinations of parameters that match observations.

The fact that fine-tuning is required is considered problematic and arguably means that our theories might be somewhat implausible; read about naturalness/fine-tuning in physics for more information.