You need to plant them. They will get nowhere in plain water. And personally I'd just sow them straight outside in the ground and not faff about trying to grow them indoors first. And good luck trying to handle wet carrot seed.

They aren't hard to germinate, they just take a while.. but they eventually come up. I've never germinated seeds like that so I can't really comment... I can say I have not had luck transplanting carrot thinnings.

Just sow them in the ground and you can always thin them later

Since they're wet and sprouted, maybe fluid seeding would be easier: http://www.coopext.colostate.edu/4DMG/VegFruit/fliud.htm