What you're talking about here is basically the same thing as a cosmic ray air shower, except that it would have to take place in intergalactic space instead of in the atmosphere, and the amount of energy involved would have to be unimaginably higher. There are two factors that make this event rather different from an air shower:
- There are fewer particles to hit in intergalactic space
- This would require the energy released from the formation of an entire galaxy to somehow be concentrated in a single particle
Reason #2 is enough, on its own, that this would never happen in practice. But since the premise of the question appears to be that, somehow, that reason has been bypassed, let me go through the relevant calculations.
First of all, the amount of energy in the particle needs to be enough to cancel out the binding energy of the galaxy and all the stars and planets inside it. From this presentation, slide 10, suppose the galaxy's gravitational binding energy is $M(10^{-3}c)^2$, which works out to roughly $10^{53}\text{ J}$ assuming $M \approx 10^{12}M_\odot$. This would be the amount of energy required to separate the galaxy into individual stars. Then, let's approximate the amount of energy required to separate all the stars, planets, etc. into atoms as $10^{42}\text{ J}$ per solar mass, which gives another $10^{54}\text{ J}$ total. So the incoming particle will have to have $10^{54}\text{ J} \approx 10^{73}\text{ eV}$ in the galaxy's rest frame. (Actually a little more because it needs to transfer some energy to the remnants of the galaxy as kinetic energy, but this excess is something like a factor of $10^{-6}$ smaller and thus negligible.)
So suppose we have a particle of energy $10^{73}\text{ eV}$ somehow propagating through the universe. Now, we know nothing about how a particle with such a tremendous amount of energy would actually interact with ordinary matter. Such a high energy is firmly into the domain of (beyond-)nstandard-model physics. For purposes of a science fiction story, you could make it do all sorts of weird things.
But, sticking to the current science for the sake of argument, let's say you naively extrapolate the known behavior of high-energy scattering to this $10^{73}\text{ eV}$ cosmic ray. The next thing to figure out is the probability of the cosmic ray scattering off the particles it meets. And the relevant parameter to characterize this is the squared center-of-mass energy, $s$. For a collision between a massive particle in motion, with mass $m_1$ and energy $E_1 = \gamma_1 m_1 c^2$, and a massive particle at rest, with mass $m_2$ and energy $E_2 = m_2 c^2$, this is
$$s = m_1^2 c^4 + m_2^2 c^4 + 2E_1 E_2$$
Alternatively, for the same massive particle and a photon which has energy $E_2$ and is approaching the moving particle at angle $\theta$ (with $\theta = 0$ being a head-on collision), assuming $E_1 \gg E_2$, the CM energy is
$$s = m_1^2 c^4\biggl(1 - \frac{E_2}{E_1}\cos\theta\biggr) + 2(1 + \cos\theta)E_1 E_2 + \text{negligible terms}$$
So $s$ for an interaction between the cosmic ray and a massive particle is a fixed, very large value. Interactions of this sort generally get less likely as $s$ increases, so a particle with $10^{73}\text{ eV}$ is basically going to pass right through matter as if it doesn't exist. But for an interaction between the ray and a photon, $s$ varies depending on the angle. It goes as low as $s = (m_1 c^2)^2$, when $\theta = \pi$ (the photon and the cosmic ray are traveling in the same direction), and goes all the way up to more than $4E_1 E_2 \approx 10^{89}\,\mathrm{eV}^2$.
This is important because the interaction between two particles is most likely at a resonance, a center-of-mass energy which corresponds to the mass of some intermediate particle. For example, the delta baryon has a mass of $1232\,\mathrm{MeV}/c^2$, and therefore interactions between charged particles and photons are particularly likely when $s = (1232\,\mathrm{MeV})^2$. The cosmic microwave background (CMB) provides an ample supply of photons traveling in all directions, and thus any charged particle with enough energy to achieve $s \ge (1232\,\mathrm{MeV})^2$ in a collision with a CMB photon will be very likely to do so quickly. Particles with such high energy simply do not propagate very far through space. This effect is known as the GZK limit, and the associated energy cutoff is on the order of $10^{19}\text{ eV}$ (in the rest frame of the CMB). The exact order of magnitude varies by the type of particle involved, but regardless of what type of particle it is, anything that has $10^{73}\text{ eV}$ will be well over it.
In fact, a high-energy cosmic ray with enough energy to destroy a galaxy will hit not only the delta resonance, but the resonances of every particle in the standard model, and any unknown particles that may exist with higher masses up to a very high threshold. (Higher than the Planck mass, thus underscoring the need for a theory beyond the standard model to explain what happens.)
Anyway, the gist is that a particle with this huge amount of energy will pretty much immediately produce a shower of other particles of all sorts, with energies rapidly dropping as the shower progresses. This is actually perfect for your scenario, because it spreads out the immense amount of energy from one particle (which, as I mentioned, passes right through matter) to a broad swath of particles which is rather well distributed for galaxy destruction.
One might then consider the question of how close to the galaxy you have to produce the highly energetic particle in order to make this work. The answer to that depends on the characteristic length scale of the shower, which in turn depends on the scattering cross section and some complicated math that I don't want to get into now. If I figure it out later, I'll come back and add details, but for now, my conclusion is that if you ignore the main reason this could never happen, it actually seems quite plausible.