No. As we've shown from the previous verses, this is an allegorical explanation of our creation in the spiritual world (Eden), and (later) our eventual fall into the physical world. These verses are rich in symbolism and allegory. Why? Because the creation of the spiritual living beings and our fall to the physical world are extremely complex and technical events. Because our minds and brains have been programmed by the physical world and its sensual events, we have little means to comprehend the technology involved within God's creation.
Let's use an example. Let's say we were trying to explain to someone living two thousand years ago - or even to a child today - how television images are broadcast from a single location into millions of households using today's technology. How would we do this? Would we give them the schematic drawings of satellites, radio transmitters, satellite dishes and television circuitry to explain how the technology works? Certainly not, because this information would be over their heads. They would not understand anything.
What we would have to do is somehow relate and compare the technology to something they had already experienced. If we were explaining this to a ten-year old child, for example, we might compare the technology to multiple people watching the same kite flying in the sky. Or we might compare it to the soldier game the child likes to play. If talking to someone living thousands of years ago, we might try to compare television and radio to one king sending out the same message using a thousand messengers on horseback going to different places.
Whatever allegorical symbolism we would use, it would be meant to try to get across the central purpose of the information exchange: What is the intended message?
As we read Genesis, we must understand the source, the intended audience, and the ultimate purpose of the discussion. In this case, this is information being handed down from spiritual teachers to their students thousands of years ago, and at some point having been put to writing. The information ultimately came from God through these teachers, and its intended purpose is to enlighten the listeners and readers about the nature of the spiritual world and how we landed here in the physical world. The only problem is that its real meaning has been shrouded by mistranslation and misinterpretation.
We can see the meaning of Genesis as we explore the symbolism coming from the Hebrew. Consider, "This is now bone of my bones and flesh of my flesh." The Hebrew being translated to "bones" is עֶצֶם ('etsem). עֶצֶם ('etsem) can be translated to "bone" but also to "essence" or "substance" according to the lexicon. In other words, the symbolic Adam is saying that he and the symbolic woman share the same essence or substance.
As for "flesh," this is taken from Hebrew בָּשָׂר (basar), which can mean the flesh of the body, or used in a more expansive manner, such as "all living things" or "mankind," according to the lexicon. In other words, instead of "This is now bone of my bones and flesh of my flesh," the message being communicated is:
"We are of the same essence - all living beings share the same essence"
And who was the symbolic Adam actually saying this to? Since he is discussing "woman" (symbolizing the community of the spiritual realm) in the third person, this would leave God. Thus, this symbolic discussion takes place between the symbolic Adam and the Supreme Being.
As for, "she shall be called 'woman,' for she was taken out of man," this phrase utilizes the same context and language discussed in Genesis 2:20, which mistakenly translated צֵלָע (tsela') to "rib" instead of "part." The Hebrew being translated to "taken out of" is לָקַח (laqach). This can certainly be translated to "taken out of," but it can also be translated to "get, fetch, lay hold of, seize, receive, acquire, buy, bring, marry" and so on. It can also be translated to "take in the hand" or "take and carry along."
The concept being communicated here is a shared essence among our spiritual community. The living beings within the community of the spiritual world are connected together. We are all made of the same essence. We carry with us a part of each other.
What is that part? It is the part connected to caring and loving, which is connected to the caring and loving of the Supreme Being. Remember in the previous verse that it was God who created both the symbolic Adam and Eve, and they shared this common characteristic or part. It was not as though Adam owned this part (or "rib") and gave it to Eve. The part comes from God. This is our nature: to love and care for God. We all share this nature, and like it, we also share a common love and care for each other - in our pure state.
This renders the meaning of the next verse: "For this reason a man will leave his father and mother and be united to his wife, and they will become one flesh."
"For this reason" is connected to the symbolic Adam and the symbolic Eve sharing this characteristic (or "part"): They are sharing this need to love and care for God and each other. And it is "for this reason" that we find the attachments of family and marriage in the physical world.
Yes, the reason why there is this community called family in the physical world - common amongst all humans, and even animals, insects, fish and plants - is because we each share this common need for our larger spiritual community of God and His children. This is why we find a glimpse of joy in caring for our spouse, our children and our parents. We all need to care for others outside of ourselves.
This is also why those of us who seek joy by selfishly consuming and striving for the respect and admiration of others are perpetually empty. We can see this clearly among famous movie stars and media personalities, who have as much money and admiration as they could possibly desire. But do these things bring them happiness? No. This is why so many stars succumb to drugs and alcohol: Despite their abundance, they are empty.
But we can also see that those stars that pursue their family life in private will find some real joy in that family life. This is because they are participating in a reflection of our spiritual community. They are caring for others outside of themselves, and this care gives them a glimpse of joy.
This joy, however, is also filled with pain because they are connecting their love and care with the physical bodies of their family, which will eventually become sick and die. In other words, all our family and loved ones will eventually leave us: if not by divorce or physical separation, then by the death of the physical body.
Our spiritual community, however, is perpetual. Our relationship with the Supreme Being is always there, and by re-connecting with that relationship, we also re-connect with our nature of loving and caring for God and the rest of our spiritual community - His children in their spiritual forms. This is our real family. And this family can never be separated from us or taken away.
And as far as Adam naming the symbolic "woman," this is a mistranslation. The Hebrew used here is קָרָא (qara') which means to "to call, call out, recite, read, cry out, proclaim" according to the lexicon, and most specifically, in this context, "to call unto" or "call (with name of God)". This, as we showed when this term was also used in Genesis 2:20, Adam is not naming things. He was not a biologist who was going around naming things.
Rather, what is being communicated is that the two of them (symbolically representing our spiritual community) shared a common nature not only to love and care for each other, but to share a relationship with and worship God - being symbolized by their "calling out to God". In other words, what is being communicated here is not "she shall be called woman, for she was taken out of man" but rather:
"we share this common essence as we call upon and worship God."
This understanding brings us to the last part of this verse: "The man and his wife were both naked, and they felt no shame." What is being communicated here?
Remember that the previous verses are discussing how we all share this nature of spiritual community, of loving and caring for God and each other. This is our pure state. This is our state within the spiritual dimension. This contrasts with the physical world, where each of us is covered up by our false identification of thinking we are these temporary physical bodies.
"Naked" is being taken from the Hebrew word עָרוֹם ('arowm), which can also be translated to "bare" according to the lexicon. In other words, Genesis is not discussing clothing here.
Consider the phrase, "to bare my soul." This is used to indicate we are revealing our inner nature to someone. This is the application of "naked" or "bare" in Genesis. The symbolic Adam and Eve (the community of the spiritual realm) were, at this point in their existence, pure spiritual living beings, who were embracing God and their spiritual community. They were in a state of being "bare" or better, "pure."
The common "part" shared by Adam and Eve reveals our pure (or "bare") identity of being God's loving servants. And it is in this state that we feel no shame, and no guilt.
Shame and guilt in this context are synonyms. What is guilt anyway? When we do something selfish we usually feel guilty or shameful. Why? Because our real nature is to love and care for the Supreme Being and His children without any self-centeredness - without expecting anything in return. This is our pure nature, and when we abandon it in favor of self-centeredness, we feel guilty because we know being selfish is not our true nature. Thus we can also say that since our essence is given to us by the Supreme Being, guilt is also a message from the Supreme Being trying to remind us of our true nature. Thus, guilt is actually God calling us back home to Him.