I think this is a case where we have to suspend disbelief to be entertained.
The first thing we have to suspend disbelief for is that they were able to perform surgery on Tony Stark's open chest in a cave in the middle of a desert. No sterilized room here! Not to mention the size of the hole in Stark's chest! See how Pepper Potts can put her hand in his chest? No idea how they could have rearranged his organs to make room!
Second, the arc reactor created is also capable of great magnitudes of energy to power an Iron Man suit, something that is pure science fiction. The chest piece is also essential because it keeps the shrapnel, or fragments, from killing Tony by entering his heart. The shrapnels must be of such a small size that even the cave desert surgery could not remove all of it.
So maybe we really need to believe that the shrapnel can truly not be removed, else Tony will die.
Well, with all this sci-fy and technology stuff, why not also believe that they would have the technology to fix Tony Stark all up when he is back in the US? Say some kind way to diffuse his blood or some other fantastic surgery to remove all the shrapnel. Very true. To which I would answer:
The chest piece also serves a purpose for the story and character. It's part of the drama that if our hero doesn't have it in place, he will die. It's a physical weakness. And who's to say that Tony doesn't want someone operating on him, when he's unconscious and vulnerable, who could then simply steal his arc reactor technology? In the screenshot above, Tony is having Ms. Potts change out his chest piece in his own lab, not a doctor in a hospital.
Or even he's not the type that wants to be fixed or cured. He is a genius, superhero, playboy billionaire with narcissistic tendencies. He may not be like us in thinking of simply getting it fixed.
It also adds emphasis to when Tony Stark says, "I am Iron Man".