Obama: If you are a US citizen the NSA can’t listen to your calls 13 years ago
The rights recognized by the Constitution are human rights. We are not granted those rights by any law. Our Constitution simply recognizes rights we inherently possess due to the fact we are human beings.
To say that other people do not possess those rights implies that the Constitution is granting US citizens those rights, not simply recognizing those rights. Its a subtle but very important difference.