Melton was injured in Sunday's win over the Washington Redskins.
Melton appeared in all 16 games for the Cowboys this season and made three starts. He had five sacks and recovered two fumbles.
The Cowboys signed veteran linebacker James Anderson to take Melton's spot on the roster. Anderson was released by the New England Patriots before the season and picked up by the Atlanta Falcons, who cut him before Week 17.
- Dan Gartland