No, our Constitution simply doesn't hold any weight in other countries. It doesn't grant foreigners rights that citizens enjoy, nor does it guarantee those rights to our citizens outside the U.S. For example, you're not going fly into Dubai and successfully invoke your 1st, 2nd, and 4th amendment rights to say what you want, when you want, while open-carrying your handgun and guarding your possessions from unreasonable search and seizure when they come for you.
If our own citizens aren't protected by the Constitution while in Syria, why would you think Syrians are protected by it?
That's debatable. The US blocked suits about Guantanamo to ensure no court case made it to the Supreme Court to rule on that.
And rightfully so. They are prisoners of war, not guests in the Hamptons. It should stay out of the civilian court system. There are military courts that are better suited to deal with cases like that, and applying more applicable laws. Such as military rules of engagement, and treaties like the Geneva Convention, etc.