Does your right to medical care from me exist outside a contract between you and I?
Rights exist because we exist. But it is the power to assert and manage those rights that determines who is sovereign over their affairs and who is a supplicant.
What I said about a state being a monopoly on violence and a government being the thing that exists when two or more people will it to be, defines the state as sovereign in this context. It will remain sovereign so long as individuals lend it their power and pay it fealty. When that ceases, then the state loses it's mandate, ceases being powerful and is no longer a monopoly on violence as the people reclaim and reassert their rights.
At a more basic level, I have the right to live, so do you. You are not a provider of medical care. Were you one, then yes my right to continue to live is going to be more important to me and mine, than your right to do with your time as you please is to you and yours.
If I can assert that right either through plea or force, then my right is recognized by you. But that doesn't mean it didn't exist prior, just that you were failing to recognize it.
The underlaying problem here is that you have paid fealty to a state which says that you are not sovereign over your financial affairs. The state has taken your money and used it to compensate medical providers to provide medical care for those who are in need.
The state does this because yes, they do have this right and it is even enumerated in some of our most basic principle founding documents. "We hold these truths to be self evident that all men are created equal and have the rights to life, liberty..."
This is a thing you disagree with, but you are not powerful enough to walk away from the society that gave you this money while still holding onto the money, power and prestige that society gave you by acknowledging your rights.
It sounds like you believe that your right to money you earned , exceeds the right to life of the indigent. Yet you choose to continue in a society that disagrees with you on this point. You have to ask yourself why you would do that, rather than asserting your rights and becoming active at changing the balance of power back to the side of the powerful from the side of the weak?
Contracts can NEVER create a RIGHT. They can enumerate the terms of agreement. Some of those terms are requirements that each side acknowledge the rights of the other.
Contracts are a reference to go off from when there is a disagreement. What you're calling a right, is just a term of agreement and it only applies so long as the parties agree to those terms.
You are correct that when parties disagree, they need an arbiter. Your system works so long as all parties agree to the terms of either the contract, or the implicit social contract which is created by participating in the society.
But rights do exist outside of that construct. It mostly matters who has the strength and determination to ensure those rights are respected.
I'm going to ask you to consider that all rights must be mutual and both parties must have the ability to grant the other party said right. If you are not a doctor then you do not have the ability to grant a mutual right of medical care.
Like I pointed out in the right to food debate: can two people who have no food enter a contract that grants each other the right to food? This this right to food exist by virtue of them being human? Do they have this right in the desert?
Would you sign a contract that grants me a right to food whether or not I provide you food or compensation? Would you open that contract up to anyone who wanted to sign it with you?