Let's address this first.
Whether we're talking about an "instinct" or a "natural right" the effect is the same. Whether you believe in God, Darwin, Gaia or, in true Epicurean fashion, that there is no truth other than that which we can validate through our senses, there IS, without question, a natural pursuit of survival and a natural process of reproduction. Whether you think of those functions as "rights" or "instinct" is immaterial to the FACT that they are part of the nature of life...all life. Humans, having more control over their survival than other species, have constructed a whole mess of facilities that improve their chance of survival. Some of those facilities are material and some are social but all of them have been created with the purpose of improving chances of survival and that is the basis of all rights.
From time to time one person or group will decide that their own survival is more important than the survival of any other person or group and that's where we run into conflict. Governments are formed, primarily, for the purpose of deciding whose rights are protected and whose are rejected. The "rights" created by government are civil rather than natural and serve only to protect or reject the fundamental (natural) rights of the people that come into contact with the government. By denying the existence of natural rights you are denying the most basic purposes of ALL life. When the basic purpose of life is denied than all manner of atrocities tend to happen. That's where the human created destruction of the environment comes from and it's where genocides come from.
With regard specifically to the matter of abortion, the idea that a pregnant woman should have exclusive and unlimited right to ALL decisions related to her unborn child MUST be tempered with the understanding that the life inside her is distinct and unique. The life inside her is human and even in the first moments after conception is pursuing independent survival which, as I have said before, is the most fundamental purpose of life and the basis for all other rights.
There is one other issue that comes up with this whole "pursuit of survival" matter and that is the issue of sacrifice. There are times that the pursuit of survival of the species comes into conflict with the pursuit of survival of the individual. We all must, at times, decide who should sacrifice when others are in need and what level of sacrifice is prudent. A pregnant woman, without question, does sacrifice for her unborn child. It's important to acknowledge that sacrifice but the mere fact that sacrifice is occurring DOES NOT mean that the rights of the person she is sacrificing for are immaterial.