There are some things you need to distinguish:
A green card expires like a driving license, you need to get a new card, but the actual right it is based on does not "expire." The same way you can continue to drive, you can continue to live in the US with the renewed document.
However, at the point of renewal the authorities will look into any reasons why someone could have lost his right to stay in the US. A criminal conviction can be a reason.
Under immigration law a criminal conviction can lead to removal. That means, the government does not have to wait until the "green card" expires, the conviction makes the alien deportable the moment of the conviction, regardless how long his green card is still valid on its face.
Once declared removable he can be ordered to leave the country or be arrested and forcibly removed.
Being married to an American citizen does not change these basic facts. Any criminal alien can be removed under the law if the conviction is basis for removability. In some cases being married to an American citizen can make a difference if the alien tries to obtain a waiver of the deportation on hardship grounds, but it is not likely for criminals to obtain such a waiver.
Now to the conviction itself: any conviction for drug offenses except a simple possession of marihuana (30g or less for own use) makes an alien removable.
