Yes, the Constitution allows a landlord not to rent to people based on their race or religion. Period.
I'll make this straightforward for you: The Constitution does not "allow" anything - - the Constitution ONLY tells GOVERNMENT those things that GOVERNMENT is allowed to do, and then for good measure goes on to specify some of the things that GOVERNMENT better not even THINK about doing. For example: "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances." (That's the 1st Amendment.)
P.S. There is a wealth of information about the basics of the US Constitution, and it is all right there at your fingertips courtesy of the internet. Happy surfing!
Regards,
LH