Initially selected on the bench for Saturday's clash with Ireland, the Gloucester back suffered the injury while warming up ahead of kick-off at the Aviva Stadium.
The 27-year-old - who can play at fly-half or inside centre - had to withdraw from the matchday squad in Dublin, Jarrod Evans taking his place among the replacements.
Wales confirmed on Tuesday that Williams will miss the rest of the tournament following further assessment, though they are yet to name a replacement.
The reigning Six Nations champions are already without fly-halves Gareth Anscombe and Rhys Patchell, while Dan Biggar was forced off during the first half of the 24-14 defeat at the weekend after failing a head injury assessment.
Wayne Pivac's side are next in action on February 22, hosting unbeaten France in Cardiff.
Photo Dan Mullan/Getty Images Caption: Owen Williams will miss the rest of the 2020 Six Nations.