Will a potato with green skin eventually ripen?

No. The potato is not green because it is unripe. Potatoes are ready to eat at any stage of growth. The green colour is indicative of poison. Potato plants contain solanine, a bitter tasting chemical. Most of this chemical is in the leaves and stems of the potato plant but smaller amounts are found under the skin of the tuber and to a lesser extent in the eyes. If you pull out a potato from the ground during its growth period it is instantly exposed to light and it starts photosynthesizing. So it manufactures chlorophyll and becomes tinged with green on the surface. 

It is always sensible to cut away and discard the green parts, the rest of the potato will be perfectly okay.