As someone who has a hobby like interest in aviation, I'm aware of quite a bit of what goes on in Greenland.
The US does have a base there, and it covers a large area, but it's not a "huge base", it's in the far north of Greenland where they have a "Space Force Base" at Pituffik (formerly Thule). It's not clear how many military personel are based there, but it's likely to be less than a couple of hundred. A lot of it's previous accomodation, built during the cold war, is likely to be out of commission these days, so would likely need major construction. It does have a near 10000 FT runway, so can take most military aircraft, and is open year round. It's basic role is early warning of ICBM, but I'm sure it eavesdrops on russia from there.
There are two other large runways, at airports called Kangerlussuaq, and Narsarsuaq, both former US military bases, but neither have any military based there now as far as I know, though the former often has USAF flying in and out of it, and sometime based there for weeks (mainly summer months), I think they retain some military buildings at both for that use.
The main airport at Nuuk (capital City) was extended about 18 months ago and can take large aircraft now, and regularly has Canadian and USAF visiting aircraft (as well as the rest of NATO), and they are building another longish runway at Ilulissat, so that it can take larger aircraft than the work horse Dash8 of Air Greenland. There are search and rescue and some transport helicopters based around the Island also, and they have daily flights to and from Denmark with an A380 airbus, as well as some chartered (tourist) smaller jets from Iceland, Greenland, and in summer Canada and the USA.
As a result of having this interest, I know there are usually a couple of Danish warships based in and around the capital on rotation from Denmark. Apart from the Danish ships, and the various military aircraft I've mentioned, I've never seen any US military there. Could the US easily take it by force, yes, but I'm not sure they would need too, because any Danish miltary there would be minimal, I would imagine a small garrison, and they probably could just waltze in any time they wanted, but they might be better just asking, and they'd likely get permission anyway.