EDITING to add: I do see this as different from choosing a physician you deal with on a regular basis, outside of the hospital. You can choose whoever you want at that point as you are not asking the hospital to discriminate on your behalf.
Sure you can, and then you be prepared to wait. I'm weird because I just want the best and fastest care, but if people are seriously freaked out about a man treating them they can wait for a woman, I guess. And if a woman isn't available (like with my son and the surgeon) you can make a choice again, And if you put your childs life at risk because of those choices I still believe that you should be culpable.
Again, you can request anything, my issue is with the hospital incorporating ILLEGAL practices at the request of a patient.
I remember this debate that was HEATED years ago about people who refused to get out of wildfire evacuation sites. I said that they assumed the risk in staying and that saving them should not be a priority, i.e the firefighters should not risk their lives saving people who had ignored MANDATORY evacuation statements. People were like, shocked at that view. I look at this the same way. You don't want to be treated by a capable person solely because of their gender? Okay. But don't complain when your treatment is 50% slower because you have opted out of 50% of the care available to you (or more).
So request away. But just like if you request morphine for fun, a hospital should not give it to you, requesting that a hospital put discriminatory practices into place should be met with a "No we don't do that" and then you can have the choice to accept it or leave. If you honestly are ill and want to deny care because of the gender of the person, fine, just accept the risk inherent with that choice. Willingly turning down available care is your choice, but a hospital does NOT have a choice legally to prohibit a nurse or Dr from working because of the color of someones skin or their sexual orientation. That is why I believe that this is a good and smart lawsuit as it is shining a light on bad business practice and active discrimination. The customer is not always right.
I still think it is a dark slippery slope to hold a parent legally culpable for requesting a different doctor or nurse, if something happens. In my opinion, if you are legally culpable, then you would also be legally culpable for requesting a different doctor because you thought they were doing a bad job. As I said before, racism is subjective.