Ferrari said on Friday that Sainz, who finished third in the Bahrain season-opener last weekend and remains the last driver to beat dominant Red Bull with victory in Singapore last September, would require surgery.
Bearman, who will make his race debut at the age of 18 years and 306 days, is reserve driver for Ferrari and Ferrari-powered Haas and would have been competing for the Prema team in Formula Two at the Jeddah Corniche circuit.
The Briton had qualified on pole position for the feature race in the feeder series, his fourth F2 pole, and said afterwards that Jeddah was one of his favourite tracks.
He is the 97th Ferrari Formula One driver and the first to make his debut with the sport's oldest, most successful and glamorous team since Italian Arturo Merzario in 1972 at the British Grand Prix.
Bearman took to the track on Friday for final practice with the number 38 on his car, a number that has featured 12 times in the past on a Ferrari and whose previous users include Britain's first champion Mike Hawthorn.
Hawthorn won his title with Ferrari in 1958.
Sainz had felt unwell on Wednesday, skipping his media duties, but took part in Thursday's two practice sessions for Saturday's race and clocked the seventh fastest time in the second one
The Spaniard is leaving Ferrari at the end of the season with Mercedes' seven-time world champion Lewis Hamilton taking his seat next year.
Ferrari did not say when he would return. The next race on the calendar is the Australian Grand Prix in two weeks' time.
Alex Albon also had appendicitis ahead of the Italian Grand Prix in 2022, which saw Nyck de Vries take his place in the Williams car, but the Thai driver returned for the next race in Singapore.
