Amendment 10 - Powers of the States and People. Ratified 12/15/1791. Note
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
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The Tenth Amendment only grants States or the people unspecified powers (not rights), and States are only granted powers not "prohibited by [the US Constitution] to the States." Several other amendments provide strong arguments against State prohibition of birth control.
The Ninth Amendment makes it clear that the people are presumed by the framers as having protected rights not specified in the Constitution. Arguably, reproductive freedom is one of those presumed rights since there were no State restrictions in place to my knowledge when the Bill of Rights was passed. Other amendments which suuport an argument against a state power to ban birth control are the First Amendment's free exercise of religion clause, the Fourth Amendment's prohibition of unreasonable searches and seizures, and the Fifth Amendment's due process provisions, all of which are applicable to the States under the Fourteenth Amendment.
Santorum has argued that states have a "right" to ban contraceptives, so your 10th amendment "powers" argument does not support Santorum's States' rights argument. Apparently Santorum never learned the difference between rights and powers in law school, which is troubling in itself.
I would expect a politician like Santorum with a law degree to have a better working knowledge of the Constitution by now. Keep in mind that anyone who recognizes a State's right to ban contraception is also suggesting that States have a right to require contraception. Suppose a State passes a law requiring all married females under 25 to use contraception based on their findings that younger women are not sufficiently prepared for the demands of motherhood. Santorum's interpretation of the Constitition would support such forced contraception.