Is it permissible for the doctor to refuse the apology and demand accountability before Allah, and will Allah punish the student for what happened, and is it permissible for the student to lie to solve the problem?

There is no harm in lying for the sake of reconciliation, such as your saying that you love her. We have clarified that whoever apologizes to his brother has fulfilled his obligation; so repeat the apology and involve some colleagues. If she insists, then there is nothing upon you.