Is it permitted to walk in a public domain on Shabbos with a NuvaRing?
Yes it is permitted. The reason is because it is similar to a ring on a woman’s finger, that it is kept there, therefore it is not considered carrying outside. I should be mentioned thought that a woman shold not use birth control without the consent of her rov, and after it has his consent that it is indeed needed, it is also considered a tzorech haguf, because at this point this is something that she needs.
Horav S. Felder, R Matisyahu Deutsch, R’ Y. Cahen, R’ Leshinsky, R’ E. Marburger shlit”a. As a side point walking outside with a ring would not be permitted if the woman might take it off in the street to show her friends, however that fear would not apply here.