This is a/the part of the Bible (Genesis 2) people reference to support that belief:

10 And a river went out of Eden to water the garden; and from thence it was parted, and became into four heads.

11 The name of the first is Pison: that is it which compasseth the whole land of Havilah, where there is gold;

12 And the gold of that land is good: there is bdellium and the onyx stone.

13 And the name of the second river is Gihon: the same is it that compasseth the whole land of Ethiopia.

14 And the name of the third river is Hiddekel*: that is it which goeth toward the east of Assyria. And the fourth river is Euphrates.

15 And the Lord God took the man, and put him into the garden of Eden to dress it and to keep it.

*Ostensibly Tigris

There's a field that was called Gu-edin (meaning "open fields") in the mid third millennium BCE that was the subject of a border war that lasted a couple centuries, between the cities of Lagash and Umma (which is right where you said), because the founder of Lagash bought an unassuming piece of land from Umma and a bunch of surrounding terrains, and then did mad irrigation work and it became crazy fertile. According to Lagash's records, Umma got mad that it was swindled out of such great land and kept attacking Lagash over it, and kept getting its ass kicked and its kings killed. People from Umma were "allowed" to till the field for Lagash for a time, but most of the grain would still go to Lagash, causing more revolts from Umma (and more punishment).

It's fairly agreed that this place probably gave some degree of inspiration for "Eden", along with some rare green gardens in the region created with irrigation work. The apple bit, the woman rib bit, and the knowledge bit came from other Sumerian myths.

I'm not sure if it's the Galapagos, maybe in the Canaries instead?, but some island famous for its apples, weather, and safety did play a part in inspiring the myth of Avalon, the island of apples.

I just think it'd be funny to imagine that the legendary first home of humanity was somewhere in southern Iraq or Kuwait

No one tell this guy about 6th grade geography.

The Galapagos weren't known to Christians until the mid 16th c. so there's a bit of a timing problem of over a couple thousand years.

Were there people exploring the Galapagos in 1000 BCE?

You do know that Genesis wasn’t written by Christians do you

Charles Darwin did not create the idea of social Darwinism, though. He suggested the biological evolution of species over time. People often pair the two ideas together because his last name is part of the word “Darwinism”.