When I enter the US, should I show my old or new passport to the US immigration? Or does the US immigration only look at my US green card?

Because you have been issued a new passport, your previous passport is no longer valid. When your consulate issues you a new passport, they should do something to the old passport to indicate this:

• keep it and not return it to you
• clip a corner of the cover
• punch holes in it
• stamp it as "invalid" somewhere
• give you a "cancelled" sticker to put on the old passport

Always use your newest passport to enter any country. It may be worthwhile to carry your old passport until its original expiration date, in case the question ever comes up of some document linking to a passport number.

A second valid passport may be issued under some circumstances. However, there is generally a special application process for this, and you would probably know whether you had done this. In the case of France:

En principe, vous ne pouvez pas avoir plusieurs passeports français. Toutefois, un second passeport peut exceptionnellement être délivré dans 2 situations : si votre passeport est immobilisé pendant une période de voyage ou si le passeport risque de faire apparaître des destinations incompatibles. ... Attention: la délivrance d'un second passeport n'est pas un droit, mais une faculté et c'est l'administration qui fixe ses critères d'attribution.

In general, you cannot have multiple French passports. However, in rare cases a second passport can be obtained in 2 situations: if your passport is unavailable during a period of travel [e.g. being held by a consulate for a different visa–ed.], or if the passport will show incompatible destinations [e.g. Israel & Iran—ed.] Note: obtaining a second passport is not a right, but a privilege and the administration sets the criteria for granting it.

The US governmental websites are contradicting themselves on whether presenting a passport is required.

https://help.cbp.gov/s/article/Article-820?language=en_US (mirror) says no:

If you are a green card holder and you do not stay outside the U.S. for 1 year or more, you should have either your green card (I-551) or your returning resident visa to re-enter the United States. You are not required to present your unexpired passport, however it is not a bad idea to carry it with you.

https://www.usa.gov/enter-us#item-34787 (mirror) says yes (if arriving from a country that isn't Canada, Mexico, the Caribbean, or Bermuda):

Arrival From Other Countries: All travelers entering the United States from all other countries need a passport upon arrival (regardless of their country of citizenship). Permanent residents and foreign nationals may also need a U.S. visa. You must apply for a visa before you start your trip.

Unless someone here can provide a clear, authoritative answer, I recommend taking the new passport to be on the safe side and also to try to appease over-zealous airline employees (obviously take your old passport too if your valid US visa is on it, but the OP has a US permanent residency so no visa).