It is true that the rainbow is the sign of God's covenant with man and with all the creatures of the earth never to again destroy the earth with a flood.

However it's important to point out that the "Noahic Law" instituted in this episode included specifically (and mainly) a command to enforce against MURDER.

"If man sheds blood, then by man must his blood also be shed."

There is relevance to the abortion issue. Here is the relevant scripture, from Genesis 9:

Surely I will require the life of any man or beast by whose hand your lifeblood is shed. I will demand an accounting from anyone who takes the life of his fellow man: Whoever sheds the blood of man, by man his blood will be shed; for in His own image God has made mankind...

Then God said to Noah and his sons with him, “Behold, I now establish My covenant with you and your descendants after you, and with every living creature that was with you—the birds, the livestock, and every beast of the earth—every living thing that came out of the ark. And I establish My covenant with you: Never again will all life be cut off by the waters of a flood; never again will there be a flood to destroy the earth.”

And God said, “This is the sign of the covenant I am making between Me and you and every living creature with you, a covenant for all generations to come: I have set My rainbow in the clouds, and it will be a sign of the covenant between Me and the earth.

There is also a relevant scripture in Numbers 35:

Do not pollute the land where you live, for bloodshed pollutes the land, and no atonement can be made for the land on which the blood is shed, except by the blood of the one who shed it.