No. That's because the IRS and state tax departments each maintain their own, separate tax collection systems.
This means you can't apply your federal refund to pay off your state taxes, nor the other way around.
Likewise, if you're filing more than one state return, you can't use the refund from one state to pay off another state.
If you're using the Refund Processing Service, you can pay your TurboTax state preparation fee with your federal refund – just not your state tax bill.