You may be misunderstanding the argument against artificial birth control. The argument against birth control is different than the argument against abortion, but is equally straight forward.
Health consists in the proper operation of the body. It is unhealthy, unnatural and in most cases immoral to diminish the health of one's body. Only an evil or insane person would purposely diminish the operation of his bodily organs and systems.
The argument seems trivial. No one would argue for the right to blind oneself or for the right to amputate one's limbs. Yet many people find it morally acceptable to completely prevent the proper operation of their reproductive system.
(Preventing the proper operation of a bodily system is morally permissible when it occurs indirectly as when, for example, medicine with harmful side-effects is used to fight a disease. This is a case of the principle of double-effect. The primary effect of the medicine is prevention of disease. The secondary effect is harm to another bodily system.)
On the other hand, there are times within the woman's monthly cycle where she is infertile. It is morally permissible to engage in intercourse during these periods to avoid the possibility of pregnancy. (The same God created both fertile and infertile periods in a woman's monthly cycle.)
Nevertheless, the most basic purpose of the human reproductive system is reproduction. And reproduction is one of the two most basic purposes of marriage. So couples should only employ "natural" birth control for grave reasons, i.e. when effects resulting from a pregnancy would present grave dangers to the family and marriage itself.