You should have really should have posted here or on ANY of computer sites/forums before buying your upgrades.
1. You should / might be able to play those games on Medium. High - I don't think so. The HD 5670 is almost an entry-level card (mainstream-ish).
2. That "600W" is literally junk. If you look at the detailed specifications, it can only provide a maximum of 264W from the +12v power rails. Everything else is on the +5v, +3.3v that modern computers don't use as much now. It is at best a "300W."
Compare the specifications of your Eagle Tech 600W to a known, high quality brand like Corsair, Antec, XFX, or Seasonic. Look at the labels. True 600W PSus should be able to provide 600W or close to 600W of power from its +12v.
Corsair G600. 576W. http://www.newegg.com/Product/Product.aspx?Item=N82E16817139023
Antec TruePower New TP-550 550W. 540W from the +12v rails (45A x 12v = 540W). http://www.newegg.com/Product/Product.aspx?Item=N82E16817371020
SeaSonic S12II 520 Bronze 520W. 480W from the two +12v power rails. http://www.newegg.com/Product/Product.aspx?Item=N82E16817151094
If you want to run the list games, and any of the more demanding and/or upcoming games on High, you'll need to spend a little moreRadeon HD 6850 or 6870 or better.
If you do, you need to replace that "600W" power supply...unless you are willilng to put your entire PC on the line. I doubt that power supply has any over-current, over-voltage, under-voltage protection...and probably lacks the recommended standard circuitry requirements.
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