Both of these debates are more accurately described as clashes between real rights and fake rights. To put it more politely, they pit negative liberty, which requires freedom from external restraint, against positive liberty, which imposes demands on other people's resources. Under the latter vision, giving freedom to one person requires taking it away from another.
But does the government have a compelling interest in forcing people to accept behavior that violates their deeply held beliefs? There is an important difference between requiring the government to treat gay and straight couples the same and requiring private citizens to do so. One is a matter of equal treatment under the law, while the other is intolerance disguised as its opposite: intolerance of intolerance, to put it charitably.
Similarly, there is an important difference between demanding that the government refrain from interfering with people's reproductive choices and demanding that business owners subsidize them. Just as no one has a right to pictures taken by an unwilling photographer, no one has a right to an IUD or a Plan B pill purchased with the money of people who do not want to pay for it.