No. I wish!! The only private school "tuition" you may be able use as a credit on your income taxes is when it's actually child care (child and dependent care credit is what the IRS calls it), like in preschool/daycare, not for grades K - 12. There are other requirements, like generally both parents must be working, and there is a limit. Before and after care at a private school may be considered child care and may qualify for the credit, up to a certain age (13?), but not the actual educational tuition.
At times, I believe some states have had some sort of a deduction for private school, so it's worth looking at your state's tax guidelines.
Donations to many private schools may be tax deductible, that has been the case at both of my son's private schools. So it's always good to donate so the school doesn't raise tuition as much, which is not deductible.