Speaking before the industry-wide Paris Air Show, Boeing CEO Dennis Muilenburg told reporters he promised transparency as the aircraft maker works to get the grounded plane back in flight.
He added that Boeing's communication with regulators, customers and the public "was not consistent. And that's unacceptable."
Later Kevin McAllister, Boeing's commercial aircraft CEO, apologised during the air show to the families of the victims who died in the two crashes.
Ethiopian Airlines plane crash
He said the firm was "very sorry for the loss of lives" after a total of 346 people were killed in the Lion Air crash last October and the Ethiopian Airlines disaster in March, both involving Boeing's 737 Max aircraft.
Mr McAllister also said he was sorry for the disruption to airlines from the subsequent grounding of all Max planes worldwide, and to their passengers who are planning to travel over the summer.
He stressed that Boeing is working hard to learn from what went wrong, but would not say when the plane could fly again.
Other Boeing executives also stressed the company's focus on safety and condolences to victims' families.
They joined the world's aviation elite and French president Emmanuel Macron at the Paris Air Show with safety concerns on many minds after the two 737 Max crashes.
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Its rival Airbus is expecting some big orders despite a slow sales year so far, and is likely to unveil its long-range A320 XLR.
The event will also showcase electric planes, pilotless air taxis and other cutting-edge technology.
The US Federal Aviation Administration has faulted Boeing for not telling regulators for more than a year that a safety indicator in the cockpit of the top-selling plane didn't work as intended.
Boeing and the FAA have said the warning light wasn't critical for flight safety.
It is not clear whether either crash could have been prevented if the cockpit alert had been working properly.
Boeing says all its planes, including the Max, give pilots all the flight information - including speed, altitude and engine performance - that they need to fly safely.
But the botched communication has eroded trust in Boeing as the company struggles to rebound from the passenger jet crashes in Indonesia and Ethiopia.
"We clearly had a mistake in the implementation of the alert," Mr Muilenburg said.
Pilots also have expressed anger that Boeing did not inform them about the new software that's been implicated in the fatal crashes.
Mr Muilenburg expressed confidence that the Boeing 737 Max would be cleared to fly again later this year by US and all other global regulators.
"We will take the time necessary" to ensure the Max is safe, he said.
The model has been grounded worldwide for three months, and regulators need to approve Boeing's long-awaited fix to the software before it can return to the skies.
Mr Muilenburg called the crashes of the Lion Air and Ethiopian Airlines jets a "defining moment" for Boeing, but said he thinks the result will be a "better and stronger company."