Does private school tuition count as child care

No.

 Tuition for children under kindergarten age can be used for the dependent care credit, as long as the child was in school so that you (and your spouse if married) could work, even if the expense is also educational. 

 Tuition for children kindergarten and higher is not eligible for the dependent care credit. However, if you paid extra for before or after school care, so that you (and your spouse if married) could work, and if that cost is itemized separately on your bill, then you can use that extra cost for after school or before school care for the dependent care credit. 

Agree that pre-K would qualify.  Just typically don't associate private school tuition with pre-K but clearly counts.

In the past, people have asked about Montessori, for example.