Canucks general manager Mike Gillis made the announcement via his personal Twitter account.
"Results of Ballard's MRI show a mild knee sprain. He will rehab with our training and medical staff. He is expected to be out 3-4 weeks," Gillis wrote.
Ballard suffered the injury early in Monday night's 4-2 win over the Ottawa Senators.
Ballard's right leg got pinned dangerously under his body as he fell backward to the ice after being hit from behind by Senators forward Milan Michalek just 2:23 into the game.
His ankle ended up facing the wrong way as he went down on top of it, and he needed help getting up and off the ice. He wasn't putting any weight on his right leg as he hopped to the locker room.
Ballard, 28, was acquired from Florida as the key part of a five-player trade at last summer's draft, but hasn't made the expected impact for league-leading Vancouver. He has five points in 42 games and is averaging just 16 minutes of ice time, playing mostly on the Canucks' third pairing.
Vancouver is already missing top defenseman Alex Edler, who had back surgery last week. And rookie Lee Sweatt, who had a goal and assist in his first three NHL games, broke his foot in practice Saturday and is out six to eight weeks.
Defenseman Andrew Alberts returned Monday after missing eight games with a shoulder injury, and veteran Sami Salo is expected back from a conditioning stint in the AHL later this week after missing the entire season with a torn Achilles tendon.
Vancouver also has rookie Chris Tanev, who has been impressive in his first eight NHL games but was a healthy scratch Monday with Alberts back.
Information from the Associated Press was used in this report.