In the US, a finding of not guilty almost always immediately ends the trial process due to the prohibition of double jeopardy (trying someone for the same crime more than once). A person who is convicted can appeal to higher courts[1] who generally rule only on the law and not on the facts. Should the appeals court find in favor of the defendant, the case will be handed back down for either retrial (if going to the trial court) or rehearing (if going to an appellate court). If in the latter case, it can be appealed up again or sent back down to the trial court. Retrial can be for either determination of guilt or for sentencing, depending on what is being appealed.
Getting a not-guilty verdict set aside is almost impossible and hinges on the prosecutor proving malfeasance on the part of the judge or jury, a rare event. Even then, reinstatement of charges is not guaranteed.
In certain cases, a not-guilty verdict under state charges has resulted in federal charges being brought, perhaps most famously in the Rodney King case. There has long been a great deal of controversy over this. Some claim that it falls under double jeopardy because the defendant is facing trial twice for the same act, but others (including the Supreme Court) believe that it's not as the federal charges are generally of violating civil rights and not for the action itself (i.e., murder or assault).
Anyway, based on this, I would expect the US to oppose an extradition request based on US law. While Knox might have been extradited had she made her way to the US before the trial began in Italy, since she was found not guilty once, she would be considered untriable in the US and therefore extradition could be blocked.
[1] This sequence is generally to an appeals court of three judges, occasionally an en banc appeal to a hearing of as many as 11 judges, and then the Supreme Court. The process is effectively the same with both state and federal trials, except that those appealing a state conviction may also try to appeal through federal appeals courts. This rarely works, though.