The ACA directly violates the right to freedom of religion by requiring coverage for this. If you obtain health care through your employer, you are subject to what your employer chooses for you. If you work for a faith based organization and dont like the coverage, get it on the open market. You cannot force people to act against their beliefs.
Yes, the ACA can violate the first amendment if it requires businesses to provide care that goes against their religious beliefs. Religious beliefs are covered by the first amendment. Forcing businesses to violate their beliefs in order to comply with a government regulation is an infringement of that right regardless of intent.
Yes, the ACA would violate the First Amendment if it requires that the owners of business provide health insurance to their employees, even if their religion prohibits birth control, because the First Amendment affords the people right to practice their religion. Their religion existed first. They cannot invalidate the major parts of the religion of many people.
I do not believe the ACA violates the First Amendment rights when it requires the owners of a business to provide health insurance to their employees, even if that insurance coverage allows for birth control. I think business owners use this argument as an excuse to avoid providing expensive health care. I think these people also need to be reminded that what they believe and the beliefs they have should not be projected onto others in the community. Their insistence for their first amendment rights are blocking other peoples rights and that is not fair.
Though the owners may have particular religious beliefs, unless the business or organization is explicitly religious (owned by religious groups or having a religious purpose) than there is no reason this violates their practice of religion. There is no mandate that the providers use the coverage and the employees should not be subject to the religious beliefs of the employers that they may not necessarily share, that would result in workplace religious discrimination.
Apart from most people using it as an argument against covering contraception (which some forms have legitimate medical uses for women's health issues, unrelated to prevention of pregnancy), it's apparently perfectly acceptable for insurance coverage to cover Viagra and other impotency-related products.
That said, it could be interpreted so broadly and so vaguely as for someone to say it violates their religious beliefs in using the power of prayer to heal sickness rather than healthcare