I'm in agreement with most of the above comments. The law protects freedom of religion and prohibits discrimination based on religion in places of employment. However, someone must be available to do the job in question, whether it's this woman or someone else.
I started working as a registered nurse when I was a Witness. Because of the JW stand on blood, I didn't believe that I could administer a blood transfusion in good conscience. I discussed this with my manager when I interviewed for my position. She was willing to accommodate my refusal to give blood by having another nurse do so, but she also mentioned that I would probably never be able to work in critical care because of my beliefs. I understood going in that my refusal to give blood as a nurse would greatly limit the types of jobs I could take. Departments like OR, ER, and ICU would probably not have hired me, not because they were discriminating against my religious beliefs, but because my beliefs prevented me from performing a frequent and necessary job.
Another similar situation arose in my job when I was a JW. I wanted to work in a hospital, as opposed to a nursing home or doctor's office, but I was pretty much limited to med/surg, where blood transfusions were only given occasionally. Growing bored with med/surg, I trained and was certified as a SANE (sexual assault nurse examiner), and took call several times a month. My job as a SANE was to go to the ER if there was a rape or sexual assault and do the rape exam. After performing the exam, I had to educate the woman (often teens and young adults) on the potential for sexually transmitted diseases and pregnancy and then offer the woman medication: antibiotics and a pill that would stop a pregnancy had one resulted from the rape.
Due to my belief that abortion was wrong at any point after conception, I felt that administering that pill was putting the "murder weapon" into the woman's hands. I gave the matter a lot of thought and prayer, and concluded that I couldn't involve myself in that role, and that if I had to offer that medication to a woman I would not be able to work as a SANE. The problem was, there were very few nurses qualified to work in that capacity. They needed me. I talked to my supervisor about it, and to my surprise she was very sympathetic. She said that she felt the same way and wouldn't offer it either because of her Catholic beliefs. She also told me that several other nurses and some of the doctors held the same beliefs. There always had to be someone available to prescribe and administer the pill, but if the doctor or nurse caring for the patient didn't feel comfortable doing so based on their personal or religious beliefs, someone else would have to do it. She told me that as long as I did the rest of the exam, I could get another nurse in the ER to administer that medication.
Within a few months of leaving the JW religion, I took a job in critical care. Ironically, one of the first patients I had to transfuse was a 12-year-old Jehovah's Witness girl. Her parents refused the blood, but the hospital obtained consent from the state and we forced it. In my current job, I give blood so commonly, I rarely even think of the ethical dilemma I once had with it. I would also have no problem now with offering a woman a pill that might stop a pregnancy after a rape, but I no longer work as a SANE.
So my thoughts are that if this woman didn't feel that she could issue these certificates because they conflicted with her religious beliefs, she could ask her manager to accommodate her. Obviously, they have to be issued. If her employer could arrange for someone else in the office to do that particular task, and it didn't create too much of a problem for other employees and didn't inhibit the ability of people to obtain the certificates, she might be able to keep her job. People’s beliefs change. The law changes. Ultimately the job has to get done and someone has to be available to do it. If her refusal to do it is creating a problem, she needs to be replaced by someone who can do it.
Cheryl