Is a Mexican passport needed to fly into Mexico from the United States?

Requirements for travel between Mexico and the US are covered by the "Western Hemisphere Travel Initiative". That website states in part that :

U.S. citizens and citizens of Canada, and Bermuda traveling by air between the U.S. and Canada, Mexico, Central and South America, the Caribbean, and Bermuda are required to present a valid passport or other WHTI-compliant documentation to enter (or depart) the U.S

One of the "other WHTI-compliant documentation" items listed is a US Permanent Resident Card (Form I-551) - or a Green Card - which you have stated you do have. So as far as leaving and returning to the US by air, you are covered without a passport.

In general Mexico also accept a US Green Card for entry, although I'm not certain that this also applies for Mexico citizens. However even if it does not, then your Mexican Voter's Card IS acceptable for entry, presuming it has a photo on it.

The only thing I'm not sure about is departing Mexico back to the US. I believe that your Green Card will also be sufficient there (it certainly would be if you were not a Mexico citizen), however I am not 100% sure of that for Mexican Citizens.

The general consensus on several travel sites like Open Travel, Answers.com for people in your situation is that you need a passport to fly.

A Mexican consulate can issue you a new passport or an emergency travel document if you cannot get your passport in time.

There are options other then flying where passport is not required for entry, such as driving or arriving by sea.

CORRECTION

As per Star Alliance guidelines it is possible to travel with Mexican Voter ID card as a passport exemption

You can also use your mexican military ID card (the card when you turn 18 you must get) if it has a photo with it. I know it can be used for land travel but I am not sure if they take it for air travel. When we crossed into Mexico last year at the border they stressed to get a mexican passport soon.

In March 2016, I flew out of Aguascalientes to Houston, TX, and I am both and American and Mexican. I did not have my Mexican passport, and that presented a problem of "overstaying" my "visa" as an American. It's impossible; I know; however, that's what happened. A lawyer had to write a document that stated that I was in violation of Mexican law, and this time I could leave without penalty or fines; however, the next time ... . You are required to have a Mexican passport to leave Mexico by air toward the United States. An INE will not suffice.

As stated in Timatic, the database used by airlines, the following Mexican documents are accepted in lieu of passports:

• national ID card
• certified copy of birth certificate
• consular registration
• certificate of Mexican nationality
• naturalization letter
• Matricula Consular
• Constancia de Identificacion
• voter's certificate with photo.

And for returning to the US, your green card is enough

So no, you do not need a passport.