Not in Cincinnati, they cannot. At least according to a jury in U.S. District court. The Archdiocese of Cincinnati fired a teacher for violation of her contract. The contract, which she apparently freely signed, required teachers to live in accordance with the teachings of the Catholic Church. Christa Dias was fired for using artificial insemination to conceive a child. The jury awarded Dias $171,000; the Archdiocese will likely appeal.
The Supreme Court, in the Hosanna-Tabor case, found that religious organizations could hire and fire their own ministerial employees based upon their own religious standards, and that the government could not force a religious organization to hire or retain any ministerial employee. Dias held that she was not a ministerial employee. This case, if appealed, will require higher courts to better define “ministerial employee.”
We should remember that Dias voluntarily sought employment with a Catholic school, and Catholic schools generally require all teachers to sign contracts which include morals clauses. We might wonder why she freely signed such a contract, and later chose to quite blatantly violate the contract, and then claim to be a victim of the same employer. We might also wonder, what right to religious freedom to the parents of the Catholic schoolchildren have? They are paying many tens of thousands of dollars for a Catholic school education, and it would seem that they have the right to expect that a Catholic school will teach and uphold Catholic Church teachings.