Size wise, there's no way of us telling if you don't tell us how much room there is in the case for the graphics card.
Socket wise, of course they fit - all graphics cards noadays use PCIe 2.0 or 3.0, and they're compatible with each other.
However, two things:
1) The 7970 requires two external power plugs - usually a 6-pin and an 8-pin, or else two 8-pins. Make sure your power supply has those.
2) Yes, the 7970 is a high-end gpu... it's the highest model that amd produces right now. It's also $400. What exactly are you trying to do with this computer? More than likely, you'd be better off spending half that and buying a 7870 XT.