George, the questions about the rights of non-citizens are in courts right now and will probably stay there for a while longer. I agree that it seems bizarre, but the detainees at Guantanomo are in exactly this "nutcracker" and therefore are in legal limbo.
To my limited understanding, part of the problem is that some folks don't understand the REAL structure of the USA. (Sadly, including some U.S. congressmen.) We are a coalition of 50 states. The EU is finally doing what we've been doing, and look how bumpy it gets when it was their INTENTION to merge. This fine point becomes operative when you talk about international treaties. Lawyers are making money hand-over-fist to argue that international treaties do not bind individual states.
The most recent example of this was when a Mexican national was put to death for a murder. The evidence wasn't in question. It was his legal status that spurred the big controversy. Apparently Texas didn't follow the rules of some U.N. sponsored treaty and let this guy talk to the Mexican Embassy to get legal assistance. Texans answered that they didn't ratify the treaty and that it only applies to federal crimes. They pointed out that USA citizens in Mexico who are stupid enough to join the drug-trafficking business don't get that particular break either. Mud-slinging, of course, has ensued.
My best answer to your question is thus that it depends on treaties and on whether the local law enforcement agencies are in a forgiving mood at the time, which is SURELY the wrong answer even if it is the technically correct one.
Brian, your counter-question is inappropriate because it misses the point of my stated restrictions. NONE of my comments apply to cases where the mother (and family) WANT the child to be born. A child born to a loving family who welcomes him/her as a blessing has a much better (though still not 100%) chance to grow up under good circumstances.
Good luck with the grandkid, by the way. I know I wouldn't trade mine for the world, but again, they were born to a situation where they were wanted and loved. Again, my comments about abortion don't apply.
To simplify it, my postion stated earlier is that a woman in a state of desperation is going to solve her problem. You can make her a criminal and force the child into a system that is overloaded and under-successful, or you can work to change the system, or you can work to prevent the desperate situation in the first place. But once the woman becomes desperate for a timely solution, you've already lost any chance to do something constructive. Therefore, don't compound the disaster that happened months earlier. Go with it and work to prevent future reoccurences.
On another comment, the "offer sterilization" approach was tried in Georgia, USA back between WW I and WW II. Eventually folks recognized that the sterilization was being applied unevenly and it became a race issue. It also failed to address the real problem, which was that men can't keep it in their pants and young girls are willing to offer them another place to keep it - for a while.
At base, this is a discussion over whether one can control a primal urge - to procreate. I believe the answer is that we will never fully succeed, though perhaps we can control the effects with birth control and sex education as a way to at least minimize the damage being done.