The 26-year-old was injured while preparing for Austria's friendly against Tunisia on Monday, with an MRI examination confirming he cannot participate in the tournament.
Baumgartner was in fine form for RB Leipzig this season, scoring 17 goals and grabbing nine assists.
"This is of course very bitter news for Christoph and for us as a team," coach Ralf Rangnick said in a statement.
"He is an important player and a central personality within our team. Now our full support is for his recovery."
However, Austria received a boost with captain David Alaba cleared to travel after the 33-year-old was cleared of a muscle injury having been substituted at halftime in the 1-0 win.
The squad will travel on June 4th. Their first group match is against Jordan on June 16th, followed by games against defending champions Argentina and Algeria.