Kyrgios then followed this up by asking for a toilet break. He took two racquets with him and smashed them in the players' tunnel before returning to the court.
After a double-fault handed Khachanov the match and the players shook hands, Kyrgios told Murphy "you're a f***ing tool bro" and appeared to spit in the direction of the umpire before he shunned the traditional handshake.
Kyrgios has been fined $A167,000 by the ATP.
"The ATP is looking further into what happened during and immediately after the match to see if additional action is warranted under the Player Major Offense section of the code.
That could result in an additional fine and/or suspension," the game's governing body said in a statement on Thursday.
Khachanov's win means he will take on Frenchman Lucas Pouille in round three, who beat Canadian Denis Shapovalov 6-4 6-4.
Kyrgios's Australian compatriot Alex de Minaur also bowed out of the Ohio tournament, losing 7-5 6-4 to Japan's Yoshihito Nishioka.