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Is it possible for a particle to decay via gravity? I know gravity is immensely weaker than the other forces, but all the other forces interact with particles. Do we need an understanding of quantum gravity to know if this is possible? Are there any theories where this is possible?

The closest I have seen is decay through virtual black holes.

This paper, for example, works out proton decay rates due to black holes if there are extra large dimensions. The paper starts by heuristically estimating the lifetime in normal space-time, estimating the lifetime as $tau sim frac1m_pleft(fracM_Plm_pright)^4 sim 10^45$ years. Since we do not have a proper quantum field theory doing the full decay calculation is not possible yet; how much to trust the heuristic argument remains to be seen. There are plenty of potential complications.

Is it possible for a particle to decay via gravity?

This is not the case in the Standard Model, which does not include gravity.

This is also not the case in General Relativity, which doesn't have a particle based mechanism for particle decay, although General Relativity does recognize that the relativistic mass of a system can change in a conversion of gravitational energy to other kinds of matter-energy, or visa versa, under certain circumstances at a macroscopic level.

It is conceivable that particle decay via gravity could happen in a realistic theory of quantum gravity. But, certainly, there is no credible experimental evidence of such a decay being observed in the real world at this time. Since that would possibly be very hard to observe, however, this isn't necessarily remarkable and doesn't necessarily imply strongly that it isn't possible.

I know gravity is immensely weaker than the other forces, but all the
other forces interact with particles. Do we need an understanding of
quantum gravity to know if this is possible?

Yes. We do need an understanding of quantum gravity to know if this is possible, since it is not otherwise possible, and understanding an inherently quantum gravitational concept requires an understanding of quantum gravity.

While there are numerous proposals for quantum gravity theories, none of them have achieved consensus support or even really credible claims that they are likely to be a correct theory of quantum gravity. So, the short answer is that we don't know and probably aren't very close to knowing.

This said, there are many qualitative assumptions that are typically made about broad general classes of quantum gravity theories that can help us to make educated guesses about what should and should not be possible by analogy to what we know about the Standard Model, about General Relativity, and about proposed quantum gravity theories.

Presumably, any decay via gravity in a quantum gravity would, like decays via other forces, have to preserve all conserved quantum numbers and conserved quantities such as electromagnetic charge, color charge, mass-energy, CPT, baryon number, lepton number, and linear and angular momentum. There is also good reason to think that gravitational decays would not violate CP symmetry.

But, it is also the case that one or more of these quantities which are conserved in the Standard Model might not be conserved in a quantum gravity theory, with mass-energy conservation, for example, being doubtful as a globally conserved quantity in a quantum gravity theory.

Are there any theories where this is possible?

Yes. There are theories where this is possible. A recent pre-print addressing this possibility states:

In extended models of gravity a non-minimal coupling to matter has been
assumed to lead to irreversible particle creation. In this paper we
challenge this assumption. We argue that a non-minimal coupling of the
matter and gravitational sectors results in a change in
particle-momentum on a cosmological timescale, irrespective of
particle creation or decay. We further argue that particle creation
or decay associated with a non-minimal coupling to gravity could only
happen as a result of significant deviations from a homogeneous
Friedmann-Robertson-Walker geometry on microscopic scales, and provide
a phenomenological description of the impact of particle creation or
decay on the cosmological evolution of the density of the matter
fields.

R.P.L. Azevedo, P.P. Avelino, "Particle creation and decay in nonminimally coupled models of gravity" (Submitted on 18 Jan 2019 (v1), last revised 29 Mar 2019 (this version, v2)) (minor punctuation and spelling corrections made editorially to abstract language quoted above without indication).

This paper concludes as follows:

In this work we challenged the assumption that the NMC between
geometry and the matter fields might be responsible for particle
creation/decay in the absence of significant perturbations to the FLRW
metric on microscopic scales. We have argued that there is only one
consistent interpretation for the modification to the evolution of the
energy density of a fluid made of soliton-like particles associated to
the the NMC between the gravitational and the matter fields in a FLRW
universe: a change in particle-momentum on a cosmological timescale
(rather than particle creation or decay). We have considered the
possibility that perturbations to the FLRW geometry on microscopic
scales, eventually in association to significant extensions to the NMC
theory of gravity studied in the present paper, may be responsible for
particle creation or decay. We have also have provided a
phenomenological description of particle creation/decay by defining an
“effective Lagrangian” which incorporates these effects.

The closest gravitation decay process we understand is Hawking radiation. A black holes with area $A=8pi M^2$ is composed of $N$ Planck masses or with $M=Nm_pl$. The emission of a quanta or a boson, so $Nrightarrow N-1$ is a sort of decay process, and will repeat until the black hole is evaporated.