Undoubtedly, the rainbow pictures God's mercy and His covenant with men as prophesied:
"Behold, the days are coming, says the Lord, when I will make a new covenant with the house of Israel and with the house of Judah--not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out the land of Egypt, My covenant which they broke, though I was a husband to them, says the Lord.
"But this is the covenant that I will make with the house of Israel after those days, says the Lord: I will put My law in their minds, and write it on their hearts; And I will be their God, and they shall be My people. " (Jeremiah 31: 31-33)
What about the colors of the rainbow? Do they have any significance? There are innumerable theories and views, some more impressive than others, but are they true? Rather than fantasize let's use the solid grounds of the Bible and be satisfied with what it reveals.
God's choice of colors is made quite evident in many Scriptures: Blue, Purple and Red (scarlet) were used in the tabernacle and in the temple later in ancient Israel's history. See Exodus 25:4; 26:1 and many other scriptures.
The color Blue is consistently the first of the three colors mentioned and is chosen as the color the Israelites were to carry as the symbol of their godly way of living. This can be found in Numbers 15:38-39:
"Speak unto the children of Israel, and bid them that they make them fringes in the borders of their garments throughout their generations, and that they put upon the fringe of the borders a ribband of blue:
"And it shall be unto you for a fringe, that ye may look upon it, and remember all the commandments of the LORD, and do them; and that ye seek not after your own heart and your own eyes . . . "
I have already mentioned three of the seven colors the human eye can detect (Blue, Purple and Red). The other four colors the human eye can see in the rainbow are Orange, Yellow, Green and Indigo. According to the scientific explanation of the rainbow colors, while it is traditionally believed that they are the seven colors mentioned above, the rainbow is actually a whole continuum of colors from Red to Violet and even beyond the colors that the eye can see. Sunlight is made up of the whole range of colors that the eye can detect.
Therefore, what our eyes can detect are not the only actual colors. This indicates that God has set colors (and symbols) from man's standpoint, and three of those are God's choice in His temple. There is no reliable indication, to my knowledge, of the symbolism of the other four colors. However, as Isaac Newton discovered in 1666, the range of the sunlight colors when combined produce the white color, indicating the sum total, completion. Significantly, this color pictures also God's truth and faithfulness:
"Then I saw heaven opened, and behold, a white horse. And He who sat on him was called Faithful and True, and in righteousness He judges and makes war." (Revelation 19:11)
Thank you for your question and for your keen interest in the Bible shown by your question.