Does Jesus teach water baptism in John chapter 3?

Does being "born of water" mean water baptism? Isolated from its context, verse 5 could be forced to suggest water baptism: "no-one can enter the kingdom of God unless he is born of water and the Spirit."

We will find the actual meaning of the verse by reading the entire passage in context.

Jn. 3:3-8 In reply Jesus declared, “I tell you the truth, no-one can see the kingdom of God unless he is born again.” 4 “How can a man be born when he is old?” Nicodemus asked. “Surely he cannot enter a second time into his mother’s womb to be born!” 5 Jesus answered, “I tell you the truth, no-one can enter the kingdom of God unless he is born of water and the Spirit. 6 Flesh gives birth to flesh, but the Spirit gives birth to spirit. 7 You should not be surprised at my saying, `You must be born again.’ 8 The wind blows wherever it pleases. You hear its sound, but you cannot tell where it comes from or where it is going. So it is with everyone born of the Spirit.” 9 “How can this be?” Nicodemus asked. 10 “You are Israel’s teacher,” said Jesus, “and do you not understand these things?

Notice what Jesus says in verse 6, as he amplifies verse 5: "Flesh gives birth to flesh (first birth), and "Spirit gives birth to spirit" (i.e., born again). Jesus is telling Nicodemus that the natural birth is is not enough to "see the kingdom of God" (vs. 5). Something else is required, to be "born of the Spirit" (vs. 8). In fact, Jesus suggests that this is simple, and that Nicodemus as a teacher of Israel should understand these things (vs. 10).

What we see here is that Jesus is speaking to arrogant attitude of the Jews regarding their status as the chosen nation. This is a critical understanding, for remember that Nicodemus was a teacher of the law, and believed that being a Jew and keeping the law was sufficient. 

Let's go to the account of John the Baptist. He is baptizing in the River Jordan when he has an encounter with the teachers of the law:

"But when he saw many of the Pharisees and Sadducees coming to where he was baptizing, he said to them: 'You brood of vipers! Who warned you to flee from the coming wrath? Produce fruit in keeping with repentance. And do not think you can say to yourselves, `We have Abraham as our father.’ I tell you that out of these stones God can raise up children for Abraham. The axe is already at the root of the trees, and every tree that does not produce good fruit will be cut down and thrown into the fire. I baptize you with water for repentance. But after me will come one who is more powerful than I, whose sandals I am not fit to carry. He will baptize you with the Holy Spirit and with fire.'" Mt. 3:7-11

Notice John the Baptist condemned the teachers of the law for their smug nationalism. Remember that John the Baptist was a forerunner (Matt 3:3). He was preparing the way by preaching water baptism for repentance. But he also knew that his baptism was inferior to the one Jesus would bring, that is, the baptism of the "Holy Spirit and with fire."

This old order, the external regulations, were passing away, and John the Baptist's ministry was transitory. It is not enough to be born a Jew. It is not enough to be baptized in water and repent. The new way cleanses the conscience. He. 9:8-10:

"The Holy Spirit was showing by this that the way into the Most Holy Place had not yet been disclosed as long as the first tabernacle was still standing. 9 This is an illustration for the present time, indicating that the gifts and sacrifices being offered were not able to clear the conscience of the worshiper. 10 They are only a matter of food and drink and various ceremonial washings — external regulations applying until the time of the new order."

Jesus is intent on telling Nicodemus that his genetics isn't enough. He wants Nicodemus to know that salvation comes in a way Nicodemus had not considered. Jesus is teaching the teacher spiritual things, the true way of salvation. Being born of the Spirit is the issue, not the water. Tit. 3:5-6:

"He saved us, not because of righteous things we had done, but because of his mercy. He saved us through the washing of rebirth and renewal by the Holy Spirit, 6 whom he poured out on us generously through Jesus Christ our Savior..."

Peter reinforces the idea of being washed through the Word via the Holy Spirit: 1Pe. 1:23:

"For you have been born again, not of perishable seed, but of imperishable, through the living and enduring word of God."

There is the natural birth, which is perishable, and the spiritual birth, which is imperishable. Being born again is a supernatural occurrence, here brought about by the Word, the word that washes us per Ep. 5:25:

"...Christ loved the church and gave himself up for her 26 to make her holy, cleansing her by the washing with water through the word, 27 and to present her to himself as a radiant church, without stain or wrinkle or any other blemish, but holy and blameless."

The testimony of Scripture is that we need to be washed by the Word and the Holy Spirit. Water does nothing for us. 

Jesus is not teaching Nicodemus that he had to be baptized.