When the Bible speaks of begettal and birth, in reference to salvation, the writer is using an analogy or speaking metaphorically. As there is a begettal when a child is conceived and a birth when he is delivered, so in salvation, there is a point at which faith is gendered in the heart of the sinner and a time for him to be born into the family of God.
Peter writes, "Have been begotten again, not of corruptible seed, (as when we were begotten by our earthly parents) but of incorruptible (a begettal which ends in eternal life), through the word of God...." (I Pet. 1:23). Paul's explanation is that "belief cometh by hearing and hearing by the word of Christ" (Rom.10:17). Again he says, "I have begotten you through the gospel" (I Cor. 4:15).
When true faith is kindled in the heart, it convicts the sinner of his sin and makes him both sorry and ashamed of his wrongdoing. He resolves to put this behind him. This of course is repentance, without which we cannot be saved (Lk. 13:3). The final act or step in becoming God's child is being born into His family. Thus Jesus said to Nicodemus, "Except one be born of water and the Spirit, he cannot enter into the kingdom of God" (John 3:6). As the babe emerges from his mother's womb into his new life on earth, so the believing penitent must be immersed and brought forth from the baptismal waters. Just as the newly delivered baby breathes into his lungs the life sustaining air, so when we emerge from the water we receive the gift of the Holy Spirit (Acts 2:38). The blood of Christ washes away our sins when we are baptized (Acts 22:16; I John 1:7) but the gift of the Holy Spirit is the source and sustainer of our new life in Christ. Thus in Titus 3:5 we are told that we are saved by "the washing of regeneration and renewing of the Holy Spirit, which he poured out upon us richly..." To this Paul adds, "If any man hath not the Spirit of Christ, he is none of his" (Rom. 8:9).
One reminder. When we seek to interpret analogies and metaphors of Scripture, we must only expect to find the major points of comparison between the illustration and the reality. Those who try to find some connection in every small detail soon end up with absurdities.