Of course, your request may not always be honored. Some facilities just don't have that kind of staffing. You can always call your doctor back (the doctor who ordered the test) and ask if they can recommend a different facility. They may or may not know of one. You can also ask or call around, as most radiology offices/labs will honor physician written orders for tests, even if the form says a specific place. And if you give the facility your doctor's office information, they will get the results to them.
Generally, if it's a scheduled, routine test or appointment and there's no urgency and you're in a big enough city, you should be able to find what you're looking for.
Secondly, you have to be polite. When you're at a restaurant, everyone knows, you can't upset the waiter...only God knows what they may do to your food. Medical staff would appreciate at least as much courtesy. So if you're in an ER and a male nurse comes to see you, or you're at a lab and a male tech comes to draw your blood, try not to physically convulse out of shock at their gender and make your request politely, "I don't mean to be difficult, but would it be possible to have a female provider? If that would be possible, I would appreciate the accommodation." Or something to that affect. Regardless if that's how you feel (that you're being difficult), politeness is far more likely to get you what you want (extra consideration) than barking orders for someone else.
Finally, it's just not always possible, particularly in urgent settings. In the ER or at an Urgent Care clinic, you can make your request when you check in, to have female providers, and most facilities will do their best to accommodate you. All the facilities at which I have ever studied, trained, or worked have made a sincere effort to accommodate such requests. But sometimes, there's just no one there. Then, modesty and shyness aside, you have to remind yourself that there is room in our jurisprudence (fiqh) for this situation. It is okay. It is presumably more important for you get taken care of and be able to return to your other responsibilities than it is for you receive female-only care. Better to suppress your instincts and properly allow for the provider to ask their questions and do their exam so that your treatment can proceed in a safe and efficient manner.
And God knows best.
DISCLAIMER: Information in questions, answers, and other posts on this site are for general information, and are not intended to substitute for informed professional medical advice, and do not establish a physician-patient relationship. The site is not intended or designed for EMERGENCY questions which should be directed immediately by telephone or in-person to qualified professionals.