The pair were appointed in December when Martin Allen departed to become manager of National League Eastleigh.
"Henry was instrumental in bringing through a number of our talented youngsters while he was academy manager," a club statement said.
"Everyone at the club would like to wish Henry all the best for the future."
Barnet say Newman left the club on Sunday, after winning four of 11 games in joint charge with Eames.
Barnet interim coach Henry Newman has left the League Two club, leaving Rossi Eames in sole charge at The Hive.