Dissolving sutures have been around for quite some time in some form or another.

The Roman physician Galen writes about using sutures made from intestines in the 2nd century AD and while we can't be certain these are dissolving there is a high chance they were. By the 10th century though the "catgut" suture was definitely in existence, these sutures were made of the fibrous parts of animal intestines (generally sheep and not cat!). Catgut sutures had strength for ~7 days allowing a wound to be closed to heal but would be broken down by the body within 90 days.

This remained the prevalent suture type until the development of artificial dissolving polymers beginning in the 1930s.