The Spaniard's absence from the second race of the season paves the way for the team's Belgian reserve Stoffel Vandoorne to make his race debut.
Alonso told reporters at the Sakhir circuit that he had done all he could to be in a condition to race on Sunday, but respected the doctors' decision.
He said he had undergone scans in Spain last week that had revealed a 'small pneumothorax' on the lung as well as rib fractures.
"Because of that...with the G-forces (there is a risk) that the fracture could move into the lung," he explained.
The governing International Automobile Federation (FIA) said a repeat chest scan had been requested before the Chinese Grand Prix and the results would be analysed before allowing him to race there.
"It's not 100 percent," Alonso said of his chances of competing in the third round of the 21 race season in Shanghai on April 17.
Vandoorne, last year's GP2 champion and a rising star in the sport, is McLaren's official stand-in and was flying back from Japan where he is racing in a domestic series.
The Honda-powered team said the Belgian would be at the Bahrain circuit on Friday, in time for first practice with Alonso staying on to lend his support.
Alonso's crash in Melbourne, after colliding with Mexican Esteban Gutierrez's Haas in the March 20 season opener, provided a moment of high drama with the race red-flagged before an eventual re-start.
The Spaniard, a double world champion, was fortunate to escape serious injury, climbing out of his wrecked car without assistance and released from the medical centre after precautionary checks.
Alonso had been due to race in Bahrain with a new chassis and replacement power unit after Honda said there was little to salvage.
McLaren revealed this week that the impact in the flying crash had been sufficient to crack Alonso's moulded seat, although they played down the significance of that.
"The fact that the seat cracked but was not broken means it did its job well," a spokesman said. "It flexed helpfully, as it was designed to do, and it efficiently absorbed a lot of the energy of the accident."
Former champions McLaren, who have not won a race since 2012, are fighting back from their worst ever season and appeared to be more competitive in Australia despite failing to score points.
Alonso had missed last year's Australian Grand Prix after a heavy crash in pre-season testing in Barcelona left him suffering from concussion.
(Reporting by Alan Baldwin, Editing by Angus MacSwan)
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