Originally Posted by
drminn
You technically need to show your green card to leave the US (to prove that you entered legally), although that is not always enforced. To check in you need a passport to show that you are allowed to enter the country at your destination (Vietnam in your case), and you need to show your green card and a valid passport to enter the US on your return.
Not the case in the US. In the past, when you entered the US with a visa there was a form stapled to your passport that needed to be returned upon departure (I-94) but this was discontinued in 2013. Since then, all you need to leave the US is a valid passport and the documentation to enter your country of destination. No one has ever been technically required to show prove of entering legally to the US upon exit, as there are no immigration checks and Airlines are not required to do this.
Other countries, yes. US: No