If someone cut their finger (lo aleinu!) on Shabbos in a way that wasn't going to threaten them however is a fairly deep cut may they get stitches put in by a non-Jew on Shabbos? Theoretically they could simply put on a bandage and wait until after Shabbos. Even though they might have a lot of pain from this, not necessarily would the pain be relieved from the stitches. As well with the bandage perhaps there is not a concern for bacteria getting in now.
The shailo could be broken down by first determining if this is a problem "tefira" (sewing -- which is normally ossur on Shabbos) by a person? If it is, perhaps since it won't be there for a long time it could be there is also a heter.
Keep in mind this is all being done with a goy, the question is is this a malacha at all in regards to putting stitches on a person. Perhaps it is but only a d'rabbanan and would have more room to allow this by using a non-Jew.