The referee later acknowledged making a mistake in not penalising David Weir for bringing down Shaun Maloney in the box in the 12th minute of the Hoops' 2-1 defeat.
Mowbray claims to have shrugged off the disappointment of his first Old Firm loss as Celtic boss and insists he will not pay too much attention to the man in the middle this time around.
"With all due respect to Steve Conroy, I don't know Steve Conroy," he said.
"I don't focus on the officials in Scotland, I don't know any of them or whether they are good or bad.
"I am sure someone, (Scottish Football Association head of referee development) Mr Dallas I assume, picked Mr Conroy for the game and I'm sure he is more than capable of doing the job."
Mowbray added: "There is no benefit in being aggrieved by the game at Ibrox. I have no interest in it.
"As I said after the game, referees make decisions and sometimes they get them right and sometimes they get them wrong.
"For the integrity of the game, it is important that the referees get the big decisions right.
"But there is nobody questioning anything about it, that's football and sometimes decisions go for you.
"We got one at the weekend against Hamilton, a handball. It went for us, one of the only ones that have gone for us this year.
"But we can take confidence from the game at Ibrox. For long spells, we were the team driving forward.
"So we will take a lot of confidence and belief, while we are aware of the strengths of Rangers."
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