Because I am overweight, my doctor says removal of a stone in my right kidney is not an option. She has assisted at over 1,000 births and is currently teaching in the nurse-midwifery program at the University of Minnesota. The prevalence of urinary stones is no higher in pregnancy than in the general population. If one occurs in pregnancy, sometimes an ultrasound can be used to establish the diagnosis so you do not have to undergo an x-ray procedure. Due to the improved filtration ability and the increased amounts of water being filtered by the kidneys during pregnancy, most women do pass these stones successfully.
Treatment usually consists of hydration and pain relief.
During pregnancy it would be especially important to watch for urinary infection, collecting a urine culture during each trimester. I could find no reference to a negative effect of urinary calculi on fertility. You are correct that the medications would be contraindicated during pregnancy but it would be very rare to have to have emergency surgery for this indication during pregnancy. If it were needed, they may choose to use spinal anesthesia or, if general anesthesia were necessary, they would do it in such a way as to not endanger the baby's health. But again, they would try to avoid this option.