Baumgartner was ruled out of the tournament after sustaining a thigh injury before Monday's 1-0 friendly win over Tunisia.
The loss of the 26-year-old RB Leipzig player is seen as a major blow for Austria.
"This is of course very bitter news for Christoph and for us as a team," Rangnick said.
The squad departs for the United States on Thursday ahead of their opening Group J match against Jordan on June 16th, before facing Argentina and Algeria.