Does Fumigate deal damage?
Fumigate does no damage; it destroys. All the creatures die; Repel the Abominable does nothing to stop it. If you wanted to keep your creatures, you'd have to seek another means of preventing their destruction, such as by regenerating your creatures (e.g. Dark Dabbling) or making them indestructible (e.g. Boros Charm). For thoroughness, protection wouldn't work.

Damage can lead to a creature being destroyed, but being destroyed doesn't necessarily have to involve any damage at all.

If it did do damage, it'd say so: Wave of Reckoning and Blasphemous Act are board wipes that deal damage.