It's not the case.
Here's something informative:
http://privateschool.about.com/od/financial/a/form990.htm according to this article, the law actually says:
"Federal tax laws govern the tax-exempt status of most not for profit private schools. These schools are usually classified as 501(c)(3) institutions. In order to maintain their tax-exempt status 501(c)(3) institutions must file a Federal Return Form 990 at the end of each fiscal year. In it they must prove compliance with federal laws regarding discrimination and political activity."
In other words, if you're going to get money from the government for your private school, no you can't discriminate against people. The government, which represents taxpayers of all races, religions, etc., cannot support a school which discriminates against certain races, religions, etc. You can still have your school, but you won't get a special tax break. I think that's reasonable.