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.
Phenom II X6 1090T @ 4.1GHz FX-8350 @ 4.8GHz // ASUS Crosshair V Formula 990FX // Sapphire Radeon HD Dual-X 7970 @ 1150/1500 // Thermaltake Frio w/ push-pull (using Antec Formula 7 Nano-Diamond + Cooler Master JetFlo fans) // 8GB (2 X 4GB) G.Skill RipJawsX 2133MHz // Corsair TX850 850W // Corsair Force GT 120GB SSD (OS) // Kingston V200+ 120GB SSD // WD Caviar Black 1TB // Windows 7 Ult x64