I live in the US where I have a green card. I am in France (of which I am a citizen) and I lost my passport. Can I just use my green card to fly back to the US?
You cannot use just a green card to fly from France to the US because you cannot clear the passport control for departing passengers with the green card. For that, you'll need either a French passport (which you do not have) or a French national ID card.
Even if you have both the green card and the national ID, the airline might require you to have a passport, but TIMATIC, the database they normally use to verify immigration documents, suggests that the two documents are sufficient, so you might be in luck.
As you have lost your passport, you may want to review the relevant page at service-public.fr.
You will need either a valid passport or valid carte d'identité (CNI) to exit France.
To enter the US, the Green card is enough, as stated in Timatic, the database used by airlines:
Passport Exemptions: - Passengers with a Permanent Resident/Resident Alien Card (Form I-551).
If you have a CNI, and end up using that and the Green card, you should print the linked Timatic extract and underline the quoted section with a pen, as it is not widely known among check-in staff that green cards constitute a passport exemption for the US.
If they decide to double-check for themselves, remind them to put US in the "Alien residency" field, or the green card won't be listed.
Besides the legal requirements, you would always have to check with the airline. I know first hand of a case where a passenger was denied at check-in with a combination of documents that immigration officer had confirmed to me were acceptable.
Because of the fines they receive when passegers are rejected, airlines tend to be overzealous about this.