R Yisroel Pinchos Bodner (Halachos of refuah on Shabbos, pp. 335-37) answers your question as follows
According to many Poskim, sewing/suturing a wound on Shabbos is a possible violation of a Torah prohibition (tefira). Therefore
Accordingly, one's wound may not be sutured by a Jewish doctor unless
it is a Life-Threatening situation (for example, if waiting until
after Shabbos will pose a risk of a possible Life-Threatening
- No Risk of Life-Threatening Infection
In most situations, if a wound is only moderately deep (i.e., it does
not penetrate beyond the full thickness of skin into the fatty tissue
or muscle tissue), and if it is properly cleaned and bandaged, it is
not an emergency to have the wound sutured immediately. One can delay
suturing until after Shabbos without causing any ill effect.
- Possible Risk of Life-Threatening Infection
However, if the wound is deep (i.e., it penetrates beyond the full
thickness of skin into the fatty tissue or muscle tissue), it should
be considered a medical emergency. In such cases, waiting until after
Shabbos will pose a risk of infection which could possibly become
There are also cases, when the wound has not penetrated the full
thickness of the skin, but — due to the nature of the wound or due to
the circumstances — a doctor may feel that a delay could pose a risk of
Life-Threatening infection. In such cases of infection risk, if a
non-Jewish doctor is not available, the suturing may be performed on
Shabbos by a Jewish doctor.
If a wound appears to be gaping open, and one does not know if it
poses an infection risk, he should treat it as a potentially
Life-Threatening condition, and not hesitate because of Shabbos
IMPORTANT NOTE: If the bleeding is pulsating or does not stop within
ten minutes, it is possible that an artery has been cut and the victim
will need emergency medical assistance. One should not hesitate at all
because of Shabbos.
- To Save Victim from an Unsightly Scar
According to some Poskim, if stitches are needed to save a victim from
having an unsightly scar, and it cannot wait until after Shabbos, the
wound may be sutured by a non-Jewish doctor.