There are a number of ways to answer this question.
First, we have to realize that the idea of the survival of the fittest had not yet been put forward at the time that the Declaration was written. Charles Darwin had not yet even been born at that point. If the Americans had rebelled after Darwin’s ideas came to prominence, they would surely have addressed those ideas in some way.
Second, survival of the fittest may be the law of the rest of nature, but it cannot possibly be the law of nature for human society. There is no way that human beings could ever live together in a society if we lived by the idea of survival of the fittest. We would essentially be degraded to the position of people in Hobbes’s war of each against all. If survival of the fittest would cause the destruction of human society, it cannot be the law of nature for human beings.
Finally, we must remember that the Founders of the US were people of the Enlightenment. They believed that God had made people to be different from the animals. God had given certain rights to all people. These were the rights that were to be protected by democracy. In that way, “nature” (which we could also take to mean “God”) had set laws that valued human rights. Because nature/God made us human beings special and different from the rest of creation, it was clear that we were also meant to have laws that protected what makes us special and that do not force us to live like beasts in a survival of the fittest situation.