While interesting to constitutional law scholars and state governments, the Court's decision in Coleman v. Court of Appeals of Maryland (opinion here) does not affect FMLA claims against private employers at all. It also does not disturb the Court's prior decision in Nevada Dept. of Human Resources v. Hibbs, 538 U. S. 721 (2003), in which the Court held that FMLA claims based on leave to care for family members or baby bonding are authorized against the states.
The difference between this case and Hibbs is that the Fourteenth Amendment to the U.S. Constitution permits Congress to implement its guarantee of due process and equal protection via "appropriate legislation." In Hibbs, the Court decided that baby-bonding and family care leave are "appropriate" because Congress was concerned with sex discrimination, and that there was evidence that states were engaging in sex discrimination against parents and women caring for family members. The "self-care" provision, though, is applicable to both sexes and is intended to remedy the costs of losing a job when one is ill. That subject is not appropriate for legislation under the Fourteenth Amendment. As the plurality opinion put it:
what the family-care provisions have to support them, the self-care provision lacks, namely evidence of a pattern of state constitutional violations accompanied by a remedy drawn in narrow terms to address or prevent those violations.
There, you're all constitutional lawyers now. The decision was fractured, with only 4 justices signing the lead opinion (Kennedy, Thomas, Roberts and Alito). Four justices dissented, joining an opinion written by Justice Ginsburg (Kagan, Sotomayor, and Breyer). Justice Scalia did not join the opinion, but agreed that the FMLA's self-care section does not apply to the states. Justice Scalia's concern is that the analysis the Court uses to decide if the Eleventh Amendment bars a lawsuit against a state is mushy and should be revised.