If you are willing to tame an Argentavis do not keep on shooting it with darts or tranqs otherwise it will die. So shoot it and wait about 5 seconds before shooting it again to prevent it from dying. Do not tame an Argentavis on Carnivore Island as it will get eaten.
An argentavis can't fit through a ceiling with one tile missing, but a pteranadon or lymantria can. Build a trap with one corner ceiling missing, aggro the argentavis, fly in one side and out the missing ceiling tile, and land. Dismount, run around, and close the entry door. Tame safely at your leisure.