The two managers squared up to each other on the touchline after Wagner had sprinted onto the pitch to celebrate Michael Hefele's 89th-minute winner at the John Smith's Stadium.
Wagner faces two reprimands - one for his celebrations and another for the confrontation with Monk.
Both clubs have also been charged with failure to control their players and/or officials after the touchline spat escalated into a melee.
All parties have until 6pm (GMT) on February 10 to respond to the charges.
Meanwhile, a Huddersfield fan has been banned indefinitely from the club's ground after unfurling a Turkey flag at the same match.
Two Leeds fans were killed ahead of a UEFA Cup semi-final clash at Galatasaray in 2000.
A statement read: "Huddersfield Town takes incidents of this nature very seriously and will not tolerate this shameful behaviour under any circumstances.
"This individual is not representative of the club or its supporters."