Gaither underwent surgery on Saturday night to repair a fracture in the tibia and fibula of his right leg after the team's indoor practice facility collapsed during a severe spring storm, injuring his leg.
The team was conducting a rookie minicamp at the time the structure fell.
Rich Behm, a Cowboys scouting assistant, was paralyzed from the waist down in the collapse.
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In all, 12 people were injured in the accident.