You only need your I-551 (green card) to re-enter the US. If the green card is expired but you are still in the US legally, there is other documentation that you'll need to provide.
You should look entry requirements in Timatic, the IATA's system for documenting entry requirements. United Airlines has a nice free interface to Timatic. Make sure you specify
USA for resident country/region and
Alien Residents for passenger type. The Timatic results are all specific to the data you enter.
Since you mentioned situations where you may not need a passport to enter a country, in particular Canada, I looked up the entry requirements for a citizen of the last country in the list (as you must specify a nationality and I don't know yours) and the only exemption for having a passport is:
Passengers with a Permit to Re-Enter (Form I-327) issued by the USA.
Having a I-551 (green card) simply gives you a visa exemption:
Passengers with a Permanent Resident/Resident Alien Card (Form I-551) issued by the USA and an Electronic Travel Authorization (eTA). They are visa exempt for a maximum stay of 6 months.
Also, Canada indicates that a passport is required.
your valid passport that you used to apply for your eTA.
Regardless of what documents are requested when you actually enter Canada, or any other country, the airline should go by Timatic. If you have any trouble with airline personnel at check-in, you'll want to be able to show what the entry requirements are and that you have the appropriate documents. In this particular case, it shows that a passport is required.