Well, in this case, both sides are wrong. The apology, or lack thereof, has nothing to do with the incident at all. It is clearly political sword rattling on the part of China and the U.S. China wants to take the U.S. down a notch from their high and mighty position as the superpower of the world. The U.S. won't back down an inch, because they want to keep their high and mighty position as the superpower of the world. Rather quite ridiculous.
America has expressed 'regrets' over the death of the Chinese pilot. They will, however not apologize. To do so would be to admit that their military made a mistake. To tarnish the image of the military, in a country where image is so incredibly important (of course you'd be hard pressed to find a country where image wasn't incredibly important). But the American military is supposed to be the best in the world. At least that's what Americans believe. Mistakes are unheard of.
There is another aspect of the American refusal to issue an apology. Mr. President himself, George W. Bush. Mr. Bush is of the old school. He's a firm believer of shotgun diplomacy (is that what it's called?). He's aiming America in an inwards direction, reversing Clinton's efforts towards world peace. He feels that America, over the last decade or so, has become less gigantic in the international sphere. He wants to carry the big stick. Would Bill Clinton have apologized to China over this incident? Unlikely. But will George Bush? Unthinkable.
So, if America refuses to back down, then what? Well, China will have to back down, and eventually they probably will. They don't have the big guns (figuratively speaking) that America does. They are in a weaker position than America, and they know it. In addition, they are too fragmented. Different people in senior levels of the government are saying different things. They cannot even achieve consensus among one another. The only thing that they all agree upon is that China was right, and America wrong. After a sufficient amount of sword rattling, they will let the American air crew go, with some sort of magnanimous gesture. But do they even have the right to hold them? The incident happened over international waters. In effect, China is holding the Americans hostage, are they not? If the incident happens in international waters, shouldn't the participants be open to any interested countries? I'm not too sure about international law.
So, I've said that probably eventually, the Chinese will back down. But I don't think that's the best solution. The best solution would be for America to make a scapegoat of the pilot of their navy airplane. If you go by what the Chinese say (and quite frankly, I find their version easier to swallow than the American one), then the pilot of the American navy plane did indeed cause the crash. Or was it that the plane had wandered into Chinese airspace unauthorized? Whatever. Anyways, America should make a scapegoat out of that pilot, blame it all on him, give him over to the Chinese to execute or whatever, and then everybody will be happy. Yes? Well, on second thought, this may not be such a good idea. Not that it'll ever work anyways. The Americans aren't ones to let "our boys" be abandoned overseas in hostile territories. At least recently they haven't. How many POW's were abandoned in Vietname, Japan, and Germany? What a short memory the American public has...