Left-back Schmelzer sustained the injury in Dortmund's 2-1 UEFA Champions League last 16 defeat to Zenit on Wednesday.
Dortmund qualified for the competition's last eight despite the defeat, progressing 5-4 on aggregate.
But they will now be without Schmelzer for the next four weeks, a club statement confirmed on Thursday.
That means Schmelzer will be missing as Dortmund continue their battle with Schalke and Bayer Leverkusen for second position on the Bundesliga table, behind runaway leaders Bayern Munich, and will be forced to sit out the Champions League quarter-finals.
Dortmund play local rivals Schalke on Tuesday, but face Hannover first on Saturday.
Dortmund's website said Schmelzer will be targeting a return for their DFB-Pokal semi-final on April 15 against Wolfsburg.
