Is there a doctrine or Biblical justification that explains why God chose the number 7 to be so significant?

There is far too much detail to dig into, and you could probably spend a lifetime concocting new correlations and patterns, but I think it's interesting to note that the ancients paid a lot of attention to the skies and there were seven celestial objects which they were familiar with that did not behave as regular stars.

http://en.wikipedia.org/wiki/Planetary_hours

http://en.wikipedia.org/wiki/Naked_eye_planets

http://en.wikipedia.org/wiki/Babylonian_astrology

http://en.wikipedia.org/wiki/Hebrew_astronomy

Contrary to that linked answer on the Judaism site, I do not believe that 7 is the natural cycle for anything. Can anyone name a single natural phenomenon that cycles over 7 days/months/years/anything? Neither lunar nor solar cycles divide evenly into 7-day weeks, there's always some leftover no matter how perfect we think 7 is.

Alright we are truly careening into wild specualtion with the question and my answer but for the sheer fun of it, I would suggest this as a possible explanation.

If we are allowed to make meanings from two numbers added, I would say seven, meaning 'totality' arises from joining 'God' (3) and the world (4) into our daily lives (3+4=7). This gives us totality on the earth. This sort of 'heavenly' plus 'earthly' number, making a total of seven would then seem to be confirmed in that the four phases of the moon casting its light on the earth occurs over a seven day period.

Seven just means then that God would like to be in our life to make it complete so that we rest in him on the Sabbath.

Do I believe what I just wrote? Just a very, very little bit.

I don't think you'll find dogma from any major tradition insisting that God pre-ordained the number 7, that it's integral to His nature, or anything of the sort. In fact, scripture suggests the contrary. Consider first God's consecration of 7 as the symbol of perfection. It's the final event (or detail) of the first creation account.

Genesis 2:1-3

¹ Thus the heavens and the earth and all their array were completed. ² On the seventh day God completed the work he had been doing; he rested on the seventh day from all the work he had undertaken. ³ God blessed the seventh day and made it holy, because on it he rested from all the work he had done in creation.

So, right off the bat, regardless of whether we assume the creation story is allegorical or literal, we can already start to make a case that God consecrated the number post-creation. He works for six days, and once everything is to His liking, he declares perfection, which just to happens to be on the 7th day. Seven thereby symbolizes perfection. Nowhere else (that I can think of) does God directly reveal 7 as a symbol of perfection; it is seemingly deduced from creation, God's reflection on the 7th day that His work is perfect.

Those preferring the allegorical reading might suggest that the story takes place in 7 days because the Jews already used 7 to symbolize perfection, or because certain prevalent natural harmonies in the universe made 7 significant, etc.. Those preferring a literal reading can start to sense that God didn't pre-consecrate 7; that consecration occurs distinctly after God creates everything. And in either case, the consecration of 7 as perfection occurs at a particular point in time in the story -- in fact, it's the point in time itself, according to the story, that defines the number of perfection, which "ends up" being 7.

Now, one could (and should) object here and suggest that, since God is omniscient and omnipotent, He could simply have planned to make all things in 6 days and declare perfection on the 7th. He was eternally aware of 7's perfection. And that's a fair point.

Thus, we turn to Jesus' own words to weigh in on our speculation. The first story in gospel of Mark wherein Jesus is challenged with a question about the Sabbath is significant. Upon being caught disrespecting Sabbath law, Jesus rebuts with a pretty revealing statement.

Mark 2:27-28

²⁷ Then he said to them, “The sabbath was made for man, not man for the sabbath. ²⁸ That is why the Son of Man is lord even of the sabbath.”

This statement points all the way back to creation. The Jews revere the number 7 as the symbol of perfection on account of God's mandate for a Sabbath. And now Jesus is telling the Jews that God made the Sabbath for them.

The point is simultaneously subtle and glaring. God doesn't need a Sabbath. And God doesn't need 7 to mean anything. We do. Therefore, for our sake, God chooses the number most beneficial and recognizable to our nature to signify perfection and denote cycles of rest and worship.

God choose seven to serve human nature, not His own. More to it, God doesn't need symbols or special numbers. We do. God just gives us what we need (or ask for) to help us stay on track.

My understanding is that seven contains a sense of completion in that it is the end of a process, while three contains a sense of completion from integration.

To give a quick illustration of three, as a person we have activity, a body, and a soul, or as the Divine has the Holy Spirit, the Son, and the the Father.

Seven as in the days of creation illustrate a process or cycle. But this cycle is made of up six days of creation plus the divine addition of rest. As taught in the Ten Commandments:

For in six days the LORD made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the LORD blessed the sabbath day, and hallowed it. (Exodus 20:11 KJV)

Here is another passages that illustrates that seven is composed of work plus the addition of one.

Here is another example of seven being composed of six plus one. And if thy brother, an Hebrew man, or an Hebrew woman, be sold unto thee, and serve thee six years; then in the seventh year thou shalt let him go free from thee. (Deuteronomy 15:12 KJV)

One can easily be established as a number indicative of divinty. The Shema (one of the most important passages is the Old Testament) indicates this.

Hear , O Israel: The LORD our God is one LORD (Deuteronomy 6:4)

So if seven is six plus one, what does six mean? We can surmise some sense of six and that it means humanity from this passage in the Book of Revelation.

Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.(Revelation 13:18 KJV)

In this context, six or 666 is used to indicate the beast as well as a person. But if seven is six plus one (the divine), then three sixs would indicate a complete separation from the divine (six without the one). Or alternatively, as far as humanity can go without joining together in life with the Divine.

Now as to the heart of your question, why is seven special? We are of course entering into speculative territory. I do not know of any Bible passage to support this directly.

My supposition rests on the idea of six plus one. Six is composed of two threes. I believe this indicates a the complete integration both inward and outward. We can integrate our hearts, mind, and action. And we can integrate our lives with our identity, the truth of Scripture, and the evidence of the World. We as created beings of the Divine have a capacity of inward and outward integration, but without the rest or grace given through the mercy of the Divine, we are always toiling to maintain ourselves in this integrity (a state of sixness). However, seven is the complete integration of a person inwardly, outwardly, and having been joined to God, which only occurs through process, eg. the days of creation.