Murray saw red after he perceived the Italian to have deliberately distracted him by shouting.
An angry confrontation followed as Murray - playing his sixth tournament since coming back from hip surgery - told Fognini to "shut up."
Responding to the row on Thursday, Fognini said: "I don't shut up."
Federer played better after every umpire argument - Zverev
He added: "We know each other really well - inside the court we are almost the same because most of the time I am complaining and he is complaining. But that is part of our job."
Fognini ended up going on to win 7-6 (7-4), 2-6, 7-6 (7-2), thanks to a decisive tie-break.
Fognini went on to play Daniil Medvedev in the quarter-finals, a match that he lost 6-3, 7-6 (7-4).
I don't celebrate, says Medvedev
Medvedev thrashes Zverev to win second Masters title