The existence of nationwide healthcare is already a violation of individual rights. Specifically, your right to the fruits of your labor (usually money) and the right to spend that money how you see fit. When you're forced through taxation or theft (but I repeat myself) to pay for someone else's healthcare, that's a violation of your property rights. Focusing on the issue of contraception is really missing the forest for the trees. There are no rights being pitted against each other, only two separate sets of rights being violated.
Also: What a way to come back to this site after almost a year. I really should learn to avoid controversial topics...
If you are saying that rights are being pitted against each other, just both being violated, then explaing what you see these rights being violated are, and the optimal solution that causes neither to be violated.
The issue here is the issue regarding birth control. That is what was discussed, and a focus around the Romney ad. There ARE other situations which end up pitting rights against each other. I could go and track those down, if you would want. Or, you can argue that rights are NEVER pitted against one another at any time, which would be an interesting argument to read.
The right's being violated are both property rights. The taxpayer has the right to decide whether or not he wants his money to go to the healthcare of others. Government-run healthcare violates this right. The Catholic Church has the right to decide where they want their money to go, and forcing them to provide contraception is a violation of that right.
The optimal solution is that the government steps out and let's people retain their property.
I would argue that rights are never pitted against each other, though to make an argument for that I'd first have to see an argument to the contrary.
* Abortion. You have the rights of a woman to her body and to decide whether to be a parent, the rights of a man to be a father if he got her pregnant, and also you can factor in the said rights of the unborn. Of course, you can end up denying rights to one of the parties here and think you work around, but you end up missing nuances by only thinking in terms of rights.
* War. Innocent civilians do die, individuals who had nothing directly to do with a possible conflict, but who are part of a nation and contribute to the events in the war by their own personal actions by being part of the nation. Like, the individuals who voted for the Nazi party and Hitler coming into power.
* The teaching of evolution in school vs parental rights. Parents are to have rights over their children, but children are to have rights to correct information.
* Individual's rights to access goods and services in markets vs the rights of shopowners to serve who they want. There are cases of racists collective belief systems that had African-Americans denied access to stores to be able to buy items they needed, or operate under the same set of rules as everyone else.
* An entire issue of community rights to social conventions people operate under vs the rights of individuals to do as they please. Included in this is public property rights and access. Take the example of Occupy. They decide they want to use a park for camping as a civil rights protest, to make a point about Wall Street corruption and income inequality as part of first amendment rights. You then face the fact the parks were not designed for such use, and the community opposed them being in there.
* Smoker rights vs the rights of non-smokers. Smokers argue they have a right to do what they want with their body. Non-smokers argue they have a right to clean air.
* Property values of home owners being reduced by what their neighbors do. In the case of a person buying property somewhere, and say the property next to it becomes a place that sells drugs or engages in prostitution, can result in one person's rights being violated at the risk of what someone else does.
Pretty much anything that isn't simply resolved comes about when you have rights in conflict. When everything is merely framed in context of rights, there is a lack of yielding or seeing a larger collective good picture. In short, you can't build an optimal ethical structure on rights alone or even one based mainly on rights.