If the Constitution included such language, immigrants would have rights they don’t have now and there would no need for immigration reform. Rather, the Constitution invests Congress with the powers to make laws and establish policies (which could do what this verse suggests if the political process leads to that end).
If the Constitution quoted Deuteronomy 17:15, the nation would need to discern somehow who God had chosen to be king. Also, in Deut. 17:20, the Bible notes that the chosen king’s descendants will rule a long time if the king follows God’s instructions. Clearly, our Constitution does not reflect those Bible verses. Furthermore, one does not need the Bible to see the reasonableness of requiring citizenship as a condition of political leadership.