The Ibrox outfit sacked the Portuguese coach on Thursday afternoon after just eight months in charge.
Caixinha oversaw a number of disappointing results including the Europa League elimination to Luxembourg part-timers Progres Niederkorn.
"The class clown. The walking talking contradiction. He had to go. Rangers board had absolutely no choice," former Celtic striker Sutton told BT Sport. "He's probably the most embarrassing manager in the history of Rangers Football Club.
"When you think about when Rangers needed stability they got a manager in, I don't know how the interview process worked but they just got a manager in who caused mayhem virtually from day one. Rangers cannot afford to let the guys who interviewed Pedro interview the new guy in the process.
"But Pedro didn't help himself, all the stupid little phrases. From we're the best team in Scotland - that was shocking - to last week when he said the only two results are win or win and they end up losing.
"How bad did it have to get? We had caravans, we had dogs, we had the Las Vegas palaver but you need to get results on the pitch and from the worst defeat in European history for Rangers to Progres to losing to Aberdeen at home for the first time in 26 years to Celtic just demolishing them under Caixinha's watch.
"You can't fall out with your own players but Pedro made the same mistake as Paul Le Guen and Barry Ferguson with Kenny Miller who has been a great Rangers player for years and it looked like he fell out with Lee Wallace as well, there was a rift in the camp."