Oh yes, I've had patients complain about me and ask to have another nurse. Not often, but it's happened. I just counted myself lucky that I no longer had to deal with them. One complained because I got her IV in one stick, but her veins rolled and so I had to dig a bit. Her complaint literally said, "Nurse should have stopped my veins from rolling." My response to my manager was, "Tell me how to do it, and I will." Stupid manager admitted that I couldn't but said I should have smiled more.
I have to confess that complaints bothered me more when we got a manager who insisted on entertaining all complaints, no matter how wild, unsubstantiated, or ridiculous. I didn't mind when old manager would call me in, roll her eyes, and say, "Consider yourself talked to, go and sin no more, yada yada." (literally her spiel for stupid complaints). New stupid manager would want to have a long discussion on why you didn't use Jedi mind tricks to control rolling veins, how you could be nicer to idiots, how we should value each and every customer (the very term is gag-worthy).
Did you act appropriately as a nurse? Did you treat your patient with respect? Can you sleep at night with clear conscience after caring for this patient? If you answered yes to these questions, then I really wouldn't sweat it.