If you observe human nature in all its aspects–physical, mental, emotional, social, what have you–then you notice that humans are naturally oriented toward certain ends. As a rule of ethics, the natural law forbids actions that are contrary to that nature or those ends. The law is “natural,” of course, because you can observe these things without the aid of divine positive law.
For example, humans, by nature, tend to form social relationships that are the foundation of societies. Stealing undermines these social relationships and causes societies to break down. It’s thus contrary to an end that human nature strives to reach, and therefore immoral.
Human sexuality, by its physical nature, is oriented toward male-female intercourse and toward procreation. (If human sexuality is oriented only toward pleasure, why is it fertile at all?) Contraception deliberately frustrates that orientation; it uses human sexuality for a purpose other than that for which it was intended, and is therefore immoral.
Most people would come to the conclusion using normal reasoning that stealing is wrong; but very few people come to the conclusion that using contraception is wrong. Does that then mean that unbelievers are not morally obligated to follow the teaching on contraception?
I think that most people end up deciding on a moral system that’s more or less based on “do no harm,” which forbids stealing much more clearly than it forbids, say, birth control or masturbation.
Contraception does not cease to be immoral simply by majority vote. On the other hand, the only way to be obligated by that particular moral law is to know that it exists. If someone doesn’t know that contraception is wrong, then that person can’t sin by contracepting, because sin requires knowledge of wrongdoing.
At the end of the day, though, sorting out who knows what and who is therefore culpable for which actions is a problem for God, not for us.