Gillespie was a fast bowler who played 71 Tests and 97 one-day internationals for Australia from 1996-2006 before going into coaching.
He was appointed last April for two years as Pakistan's Test coach but quit before a series in South Africa less than eight months later after an unsuccessful start.
He was the second foreigner to leave Pakistan's program in less than two months after South African Gary Kirsten stepped down as limited-overs coach in October.
Aaqib, Pakistan's national selector and interim head coach, acknowledged this week after Pakistan's failure to reach the knockout stage of the Champions Trophy after losses to New Zealand and India that the chopping and changing didn't help the national team.
"We have changed nearly 16 coaches and 26 selectors in the last two years or so," he was quoted as saying at a news conference to announce the latest overhaul of Pakistan's limited-overs squads.
"You put that formula on any team in the world, I think they will also be in the same situation.
"Until you get consistency right from the top down to the bottom, from the chairman down, then your team will not progress."
Gillespie responded to those comments by posting an Instagram story late Wednesday, describing the analysis as "hilarious."
"Aaqib was clearly undermining Gary and I behind the scenes campaigning to be the coach in all formats," Gillespie posted. "He is a clown."
Aaqib, a former Pakistan cricket international, had been acting as interim coach of the limited-overs squads following Kirsten's departure when he took over the Test squad from Gillespie.
Aaqib's contract initially ran until the Champions Trophy but he is set to continue in the role while the Pakistan Cricket Board searches for a full-time replacement.
