on a 32" 1080 tv
Screen size has nothing to do with how the video card will perform. The pixel size does matter, instead. No matter how big the screen is, the video card will recognize it by only the pixel size. Yes, your computer will tell you it is connected to 32" TV, but that doesn't affect the display performance.
Screen size matters to your eyes, however. I once played Batman Arkham City on 50" screen. All the performance was as smooth as on 21.5" LCD monitor, as their pixel size are the same, 1920 x 1080, but on that much big screen, I felt nausea.
For the video card, considering many reviews by professional reviewers and also by enthusiast users on various forums, I think 7970 on a single 1080 screen is overkill, if future proof is not at issue. I am playing on GTX 560 Ti 448 Core now, and it maxed out Crysis 1&2, Batman Arkham City, and Civilization V. I don't have PC version of BF3, but reviews show that they are on around 60 fps with this card. If less than $300 card can play around 60 fps today's demanding games on 1080 screen with max setting, why throw $500 for overqualified card? Again, this is from someone who is not interested in multi-display gaming.