This very instant? I suspect not well. But conceptually? Yes. If we made a space based telescope that could do IR-visible-UV spectroscopy with decent resolution/sensitivity we would be able to get some information about the chemical make up of the gasses on planets transiting their stars.
Much like the kerfluffle about phosphine on Venus, there are some compounds you just wouldn't expect from strictly geologic processes. Molecular oxygen comes to mind, but also some complicated shortish life time compounds would imply industry maybe. If we had /really/ good sensors we might even be able to spot the isotopic differences, it's been awhile since I've looked into the capacity for spectroscopy, but since the masses are different, the vibrational modes should be different, resulting in different spectra. We could conceivably spot if weird isotope distributions were in their environment, if it differed wildly enough from their suns make up you might be able to make the case that weird non-geologic nuclear reactions were taking place.
The reason the astronomy community is excited about the James Web Space Telescope is that it can do a few of these investigations.
My hypothesis is that, if given a (simulated) spectroscopical observation of Earth's atmosphere from a distance, astrophysicists would come up with plausible non-biological, non-industrial explanations for it.
Empirically, we've just had our first occurrence of astronomers not being able to explain a compound in a planet's atmosphere, and it's something much easier to deal with than a massive amount of oxygen.
I think they would settle on the opinion that any explanation would be about as interesting as life. But not fully settle on life.
Would fluorocarbons be detectable in a remote spectrogram of the atmosphere at the levels they existed in the earth's atmosphere? Would the depletion of ozone in the atmosphere be more visible?
From [0] it looks like CFC's peaked on the order of 1 ppb. Humans (perhaps) just detected phosphine at the level of ~20ppb in Venus's atmosphere through telescope spectral analysis [1], but that's on a very close planet that we've been staring at intently for 4000 years; presumably there are compounds present in smaller concentrations that we haven't seen yet. Is there a heuristic for the hypothetical minimum sensitivity relative to interstellar distances?
It's currently possible to detect the atmosphere of some exoplanets. The presence of oxygen would be a pretty good indicator that photosynthesis is happening.
Much like the kerfluffle about phosphine on Venus, there are some compounds you just wouldn't expect from strictly geologic processes. Molecular oxygen comes to mind, but also some complicated shortish life time compounds would imply industry maybe. If we had /really/ good sensors we might even be able to spot the isotopic differences, it's been awhile since I've looked into the capacity for spectroscopy, but since the masses are different, the vibrational modes should be different, resulting in different spectra. We could conceivably spot if weird isotope distributions were in their environment, if it differed wildly enough from their suns make up you might be able to make the case that weird non-geologic nuclear reactions were taking place.
The reason the astronomy community is excited about the James Web Space Telescope is that it can do a few of these investigations.