The title of the thread says it all.
If not, what proximity to a continent does an island have to be considered part of the continent?
Geographically, no–it’s Polynesia (or Oceania). Culturally it is both Polynesia (the natives) and American. Greenland is right next to Canada but is more generally considered with Europe because it is part of Denmark (until next week!). Oceania isn’t a continent, but that region is often grouped as though it were.
Geologically speaking, Hawaii is not part of any continent. Hawaii comes from a hotspot right in the middle of the Pacific plate. It has no geologic connection to the North American continent.
For an island to be considered part of a continent (geologically), it would have to be on the same tectonic plate. One example is the island of Great Britain, which is part of the Eurasian plate.
Geologically speaking, Greenland is part of North America, as it lies on the North American plate.
Hawaii is geologically not part of North America. Politically it is. Culturally it’s a hybrid, I suppose.
Greenland? Geologically part of North America. Politically it is not. Culturally a hybrid (European/Inuit).
The two places are interesting opposites.
Ethnologically, it’s part of Polynesia; politically, a state of the U.S.A.
Geomorphologically, it’s the most recent expression of the Mid-Pacific Mantle Plume Hot Spot, which has “moved” south and then east-south-east through Cenozoic time. The oldest former islands, on the north-south axis, consist of the Emperor Seamount chain. Starting at Midway, the chain turns abruptly east to comprise the extended Hawaiian chain. It is not part of any continent but a midocean hot spot.