All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, sallallaahu 'alayhi wa sallam, is His slave and Messenger.
There is no doubt that adultery is one of the major sins, but whoever repents, Allah the Almighty accepts his repentance, and the one who repents from sin is like one who has no sin. So, this man's repentance is sufficient between him and his Lord.
The carrying out of hudud (legal corporal punishments) is entrusted to the Muslim ruler, and it is not for the general public to carry them out. If there is a ruler who enforces such Hudud, this man is not religiously obligated to confess in order for the punishment to be carried out on him. Instead, he is required to conceal his sin, as stated in the Hadith narrated in Sunan al-Bayhaqi: The Prophet, sallallaahu 'alayhi wa sallam, said, “Whoever commits any of these disgraceful acts should cover himself with the veil of Allah, for whoever exposes his sin to us, we will enforce the Book of Allah upon him.”
Allah Knows Best.