You misunderstood.
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 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, having the payment taken out of a designated bank account, or you can pay directly on the IRS website. You must pay your state tax due using the state’s preferred method of receiving payment. Most states want you to either mail in a check or money order, or pay directly on the state website.