Etymonline suggests that the noun form was first noted in 1914:
c.1600, "face to face," from Fr. facial, from M.L. facialis "of the face," from facies (see face (n.)). Meaning "of the face" is from 1818. The noun meaning "beauty treatment for the face" is from 1914, American English.
While it's difficult to pinpoint the exact phrase that it abbreviates, looking at probable candidates such as facial treatment, facial pack, facial cleaning, facial cleansing, and facial massage, suggests that facial massage was the most likely. The following instances all antedate the 1914 suggested in etymonline:
From Beauty, Its Attainment and Preservation (1892):
Aside from its physically remedial effects, it has also been recently adopted as a beautifier, and facial massage is one of the expedients now resorted to by the fashionable woman in her endeavor to circumvent old Time by as far as possible ...
From The Art of Massage: A Practical Manual for the Student, the Nurse and the Practitioner by John Harvey Kellogg1 (1895):
Massage for Wrinkles — Facial massage may be made useful in removing wrinkles.
I have no idea who decided to abbreviate facial massage to facial. But the probability of it having been those involved in the industry is high.
1 Yes, the very same one