A standard vaginal exam or speculum check by your doctor doesn't open or damage a healthy cervix, even with your history of cervical insufficiency. The fact that your cervical length came back at 3.6 cm the very next day is reassuring and shows no harm was done. That length is well within the safe range at 15 weeks.
Next Steps
Keep up with your obstetrician's cervical length monitoring through the rest of this pregnancy, since that's the right way to watch for insufficiency. Do mention the white discharge separately, as it may need its own check for infection.
Health Tips
Call your obstetrician right away if you notice bleeding, a sudden gush of fluid, or pelvic pressure and cramping.