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.