What should a person do after Allah has accepted his repentance from an unlawful act, even though he did not respond to the request of someone who asked him by Allah to abandon that act?

You must repent sincerely for the unlawful acts you have committed and follow up bad deeds with good ones. Scholars have differed regarding the ruling on responding to someone who asks you by Allah. The apparent meaning of the hadith "Whoever asks you by Allah, give him" and "Cursed is he who asks by the Face of Allah, and cursed is he who is asked by the Face of Allah and does not give" indicates obligation. Ibn Uthaymeen said: If answering the one who asks entails harm, then he is not obligated to answer. As for your companion's statement "By Allah upon you" (or "I ask you by Allah"), it is not an oath, and there is no expiation for it. However, if he explicitly states an oath, such as "By Allah, you must do such and such," then an expiation for an oath is due. Ibn Taymiyyah said: "I ask you by Allah to do such and such" is a request and not an oath, and there is no expiation for this if one does not answer his request.