The Briton suffered the injury to his right wrist approaching the final 10 minutes of FP1 on Saturday morning, locking up into the opening turn, Sainte Devote.
Bird took the escape road at high speed, with the right front wheel of his McLaren hitting the barrier and with his hands still on the wheel.
Despite getting out of the car unaided and walking back to the pits, further assessment has revealed an injury.
Bird previously damaged the same wrist in London during the 2022 season which meant he missed the season finale in Seoul.
Following a trip to hospital where he was X-rayed, McLaren confirmed that Bird has broken one bone in his left hand and "will require further treatment".
A statement from the McLaren team read: "Due to a hand injury sustained during Free Practice 1, Sam Bird will not participate in the remaining sessions at the Monaco E-Prix.
"As a result, and pending approval from the FIA, Taylor Barnard will be stepping in for the remainder of the event."
Barnard is McLaren's reserve driver in Formula E and has taken part in a number of rookie sessions with the team, the most recent coming earlier this month in a 30-minute session ahead of the Misano E-Prix which he topped.
The 19-year-old Briton has yet to start a Formula E race and is currently racing in Formula 2 PHM AIX Racing, recording a best result of 13th so far.
Previously, the Aston Martin Autosport BRDC Award finalist in 2023 raced in F3 with Jenzer Motorsport last year where he finished 10th overall after recording one win at Spa.
Bird is one of several drivers who has suffered a hand injury following a crash in the last year.
Envision driver Robin Frijns missed four Formula E races last season after suffering a collision in the opening race of the season in Mexico City, with the Dutchman requiring surgery to pin and set the broken bones before returning over two months later.
In Formula 1, Daniel Ricciardo suffered a similar fate after crashing during practice for the Dutch Grand Prix in 2023 which also sidelined the Australian for four races across a two-month period.
