I completed my taxes with TurboTax and it turned out that I owe Georgia, I thought that TurboTax is going to pay the debt for me but I don't see an indication to that

[xmasbaby0]

Your state tax due cannot be paid from your federal refund.

It is not possible to pay your state tax due from your federal refund, nor is it possible to pay a federal tax due from a state refund.  The federal and state treasuries are completely separate from each other.  You may have seen a TurboTax summary of your federal and state amounts that confused you into thinking one was “paying” the other.  It is not.  If you have federal tax due you can pay by mailing your payment with the 1040V voucher, having the payment taken out of a designated bank account, or you can pay directly on the IRS website.

https://www.irs.gov/payments/direct-pay

You must pay your state tax due using the state’s preferred method of receiving payment.  For most states that will be by making a payment to the state’s own tax website, or by mailing a check or money order.

https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-pay...

[DoninGA]

In the File section of the program on Step 2, when you have state taxes owed, you are given several options on how to pay the taxes.  What did you select?

TurboTax cannot and does not pay any type of taxes owed, federal or state.

If you were give the option to have the taxes owed debited from your bank account and you selected that option, then you had to enter the bank routing and account information.  This information is sent with your state tax return and the state debits your account.