El Paso County records show 29-year-old Johnnie Lee Carter was being held on a charge of injury to a child causing serious bodily injury.
On Sept. 16, Odessa police were called to a domestic disturbance and found the girl with facial injuries and her mouth and eyes glued.
According to an affidavit, the toddler's mother told police that Carter choked and punched the girl then applied adhesive. A super glue stick was found nearby. Carter had fled.
Carter was arrested Monday at a motel in El Paso. He faces extradition to Odessa, some 270 miles away.
Online records didn't list an attorney who can speak on Carter's behalf.