So there's no reason why, as Gingrich's remarks appear to assume, an American citizen would be treated differently than a non-citizen when it comes to Miranda.
To be fair, it's not entirely clear whether Gingrich is mistaken about how the law is applied, or if he believes that non-citizens should not have the same rights as citizens. Here's the exchange from the Daily Show last night:
Gingrich: The American public doesn't understand reading Miranda rights to terrorists in Detroit when it's fairly obvious they're terrorists.
Stewart: The only thing I would say to that is didn't they do the same with Richard Reid who was the show bomber?
Gingrich: Richard Reid was an American citizen.
Reid was a British citizen, so Gingrich is wrong on the facts. But let's say Gingrich was right: is he saying that an American citizen would be advised of their Miranda rights, but a non-citizen like underwear bomber Umar Abdulmutallab would not?
Even the White House blog has pounced on the Gingrich's factual error, but doesn't note the false assumption that apparently underlies the remark.
This post on the Volokh Conspiracy law blog examines the issue in some detail. Aliens cannot "be criminally prosecuted in the civil justice system without the normal constitutional protections," writes Eugene Volokh. He quotes James Madison:
it does not follow, because aliens are not parties to the Constitution, as citizens are parties to it, that whilst they actually conform to it, they have no right to its protection. Aliens are not more parties to the laws, than they are parties to the Constitution; yet, it will not be disputed, that as they owe, on one hand, a temporary obedience, they are entitled in return to their protection and advantage.
If aliens had no rights under the Constitution, they might not only be banished, but even capitally punished, without a jury or the other incidents to a fair trial. But so far has a contrary principle been carried, in every part of the United States, that except on charges of treason, an alien has, besides all the common privileges, the special one of being tried by a jury, of which one-half may be also aliens.