The point is if we treat the Constitution as the liberals do, as a 'living document' that changes over time, we are second guessing what the writers intended rather than what they stated. It's a slippery slope, which is why may of us are originalists.
We should do EXACTLY what the Constitution says unless we wish to change it. Luckily, the founding fathers left us an avenue by which we can amend the Constitution when things like this arise. Until that time, they're legal if born here.
Whether or not the parents should be allowed to stay is a different matter. I'm a hard-liner about this and say that the baby can stay if they put it up for adoption, otherwise it goes with them back to their homeland.