I recommend Samsung Kids Pediatric and Adolescent Clinic in Gupabal Station, Eunpyeong-gu, Seoul.

This is Samsung Kids Pediatrics near Gupabal Station in Seoul.

The clinic is quite large, with only three pediatricians on staff. They're open on Sundays and public holidays. Since it's near the store, I've visited a couple of times, and each time I've been, they've always been friendly and helpful. And with only three doctors, you can quickly see a doctor without waiting, which is a plus.