The Supreme Court case here: http://www.supremecourt.gov/opinions/11pdf/10-553.pdf
The upshot:
Given this understanding of the Religion Clauses—and the absence of government employment regulation generally—it was some time before questions about government interference with a church’s ability to select its own ministers came before the courts. This Court touched upon the issue indirectly, however, in the context of disputes over church property. Our decisions in that area confirm that it is impermissible for the government to contradict a church’s determination of who can act as its ministers.
Everyone dust off their copies of Ordinatio Sacerdotalis.
Hat tip: Fr. Philip