The 46-year-old's seven-month spell in charge at Ibrox saw him win 14 of 26 games, but Rodgers suggested Caixinha should have been given longer to prove his credentials.
"I'm disappointed for Pedro," Rodgers said. "From my chats with him I knew how much the job meant to him and how much he wanted to do well there.
"It's another demonstration of modern football. Time, you just don't get," the Northern Irishman said.
"I have empathy for him and his job and I'm sure he will go on and get back into football sooner rather than later."
Celtic have won the league for the last six seasons.
Rangers, Scottish champions a record 54 times, were demoted to the fourth tier in 2012 after going into administration, returning to the top flight in 2016.
(Reporting by Shrivathsa Sridhar in Bengaluru; editing by John Stonestreet)
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