Why do we say "the right to life" but "the right to die", instead of "the right to live" or "the right to death"?
Life, as Merriam Webster has it defined is: 3. the experience of being alive (which is a noun). Right to Life therefore means to have the right/freedom to experience being alive. There are a lot of political connotations, but Right to Life negates discussions about what a legal human is and whether an embryo is technically alive. A right to live is still correct when discussing a viable creature (the mosquito lost its right to live when it flew at my face).
Die, as Merriam Webster has it defined is: 1. to stop living (which is a verb). Right to Die is having a right or freedom to stop living. However, we all have a right to death, because death is inevitable and no one can stay alive indefinitely. However we can be kept alive artificially (in some cases against our will) for a period of time, thus robbing us of the freedom to stop living on our own terms.
This answer is a bit theosophical or philosophical and traces its origin to Judeo Christian moral framework.
As in the Genesis, then God said, "Let us make man in our image, after our likeness". So God created man in his own image. No matter we like it or not, we come into being— a state of fruition where "life" already infused, God ordained to carry His Will. Hence the phrase, " Right to life".
So life is a gift. Our life is not ours to lose. We are only overseers of our lives. Death is already the cessation of life but, " Do not be a fool — why die before your time?" by choosing suicide, euthanasia or Santhara( of the Jains) or the likes. So the phrase, " Right to die".
These phrases own their origin to these matters of belief.