Nope, sorry to say, it's your TV.
Here's the deal... It doesn't matter if you have a 32" or gigantic 60" TV, if it's 1080p then they both have exactly 1920x1080 pixels. The only thing that changes with the larger TV screen is that the space BETWEEN the pixels, or the pixels themselves, get bigger. There's still the EXACT SAME number of pixels on ANY 1080p TV.
Generally speaking, this is compensated for by the fact that the bigger the screen the further away you tend to sit from it so you can take in the entire screen comfortably. If you're sitting too close to the TV, you'll see exactly what you're describing.
HDMI won't make a bit of difference in this regard. If anything, I find HDMI's colors to be a bit more washed out looking compared to other connection methods. But it won't magically fix what is essentially the fixed position and size of the pixels on your TV display panel.
So recently I got a new tv. It's a 47 inch LG LCD tv with LED backing and has a 1080p resolution. I was pretty excited because I knew that my xbox was going to look great after hearing about what HDTVs does to it. I hooked up the HD component cables that came with the xbox, switched it to HD and set up the resolution through xbox settings. My problem is though is that when I play games now the graphics look kinda bad. The edges are rough and outer textures look pixelated. I've been doing research and I've heard that and HDMI cable is required because the component cables that come with the xbox won't do the picture quality justice. Is this true? Also is there anything I might be missing?

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