No, I would not agree the two together "water and spirit" is saying Gospel and Spirit or Word and Spirit? I think Jesus said "water and Spirit" and meant "water and Spirit". Among other things, being born again is the act of having sins forgiven. That is one essential feature of being born again. We know from the Gospels that baptism for the forgiveness of sins was well known from the actions of John the Baptist. And in fact in the very next chapter of the Gospel of John, we read that Jesus was making and baptizing even more disciples than John even though it was the Jesus' disciples that was doing the baptizing.
And in the closing verses of the Gospel of Matthew, Jesus gives the command to his disciples that we now refer to as the Great Commission. In that command Jesus said "Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age." Now the point I want to make here, one that I have many times before, is that baptizing and teaching are participials modifying the command "to make". As such baptizing and teaching are not commands in themselves, but rather describe the method for making disciples. That is the grammatical fact.
Peter's command at Pentecost was not happenstance. He knew full well what he was saying. And he knew what was involved in his command to be baptized (Acts 2:38).
Now I will add here that Peter was not advocating what some refer to as baptismal regeneration. There is no indication that it is baptism that is the agent in the forgiveness of sins. However, there, and in many other passages, the message is clear that baptism is the occasion, the time in the life of the sinner, that God forgives his sins. Verse 39 declares that to be the promise to those present and to those far off. That promise being salvation from their sins. And of those present it says that about 3000 that were baptized were saved.
