No, federal taxes owed are not paid with your state tax refund. If you are getting a state refund and have a federal amount due, you would simply make the payment for your federal return and then enter your checking account number for direct deposit or choose to have a check mailed to you for the state.
If you have back due federal taxes, the IRS can levy your state refund.
At the end Turbo Tax shows you the net combined amount of federal and state. But that is only for your own info. You will get the full state refund and need to pay the federal tax due yourself.
How to pay the IRS
https://ttlc.intuit.com/community/tax-payments/help/how-can-i-pay-my-federal-taxes/01/26212
Maybe this will help. Different ways to pay the IRS
https://ttlc.intuit.com/community/tax-payments/help/how-can-i-pay-my-federal-taxes/01/26212
You can pay the IRS directly at
https://www.irs.gov/payments