In short: Yes, but you'd have to either be very lucky or be carrying out a continuous, 24/7 all-sky survey to catch it, and you would only see the chemical aftereffects of the war on the planet's spectroscopic signature, not the exchange itself (which would be too faint to be detectable at anywhere near Earthlike stockpiles of nukes). There was a study undertaken on the matter last year, along with several other apocalyptic scenarios, which was summarized at surface level by The Atlantic and is available here on arxiv.org. The relevant sections, emphasis mine:
Given that the world’s nuclear arsenal is equivalent to around 10^19 J of energy, the resulting radiation from its combined detonation would be much fainter than a typical GRB. If we assume that the energy is released on a similar timescale and with a similar spectrum to a GRB, a nuclear apocalypse is equivalent in bolometric flux to a GRB detonating around a trillion times closer than its typical distance. If we take a nearby GRB such as GRB 980425 (Galama et al 1998) which is thought to have detonated around 40 Mpc away, then we would expect a global nuclear detonation event to produce a similar amount of bolometric flux only 8 AU away!
Therefore, for us to be able to detect nuclear detonation outside the Solar system, the total energy of detonation must be at least nine orders of magnitude larger [than Earth global nuclear arsenal -- ed.], i.e. the ETIs responsible for the event must engage in massive weapon proliferation and concurrent usage. However, the production of fallout from terrestrial size payloads, which persists for much longer timescales, may make itself visible in studies of extrasolar planet atmospheres.
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Global nuclear war therefore potentially offers several spectral signatures that could be observed: a gamma flash, followed by UV/visible airglow and the depletion of ozone signatures. However, the aftermath of a global nuclear war will also act to obscure these spectral signatures. Groundburst nuclear explosions generate a significant amount of dust that will be lofted into the atmosphere. Airburst explosions do not generate dust, but still introduce particulates into the atmosphere. Atmospheric effects of nuclear warfare have been extensively modelled in climate simulations, the global consequences being known as “nuclear winter”. Recent simulations have shown that even with reduced modern nuclear arsenals severe climate effects are felt for at least ten years after a global conflict, especially due to the long lifetime of aerosols lofted into the stratosphere (Robock et al. 2007). They show that the atmospheric optical depth is increased several times for several years. The worst effects are confined to the northern hemisphere given that the model includes conflict over the US and Russia, though the entire planet is affected to a lesser extent.
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Hence, to confirm that a planet had been subject to a global nuclear catastrophe would require the observation of several independent signatures in short succession. One on its own is unlikely to be sufficient, and could easily be caused by any number of other processes on planets with potentially no biological activity whatsoever. There are cases beyond a global nuclear catastrophe that a spacefaring civilisation might be able to inflict on itself, given that the destructive energy at their disposal would be far greater than nuclear weapons (Crawford and Baxter 2015), including redirecting asteroids. These would be far more destructive than nuclear warfare but would generate observable signatures different than those of a naturally occurring impact event.
