Mr Foster, 32, was trapped when more than 1,000 tonnes of rock fell from the face of the El Capitan monolith on Wednesday.
A spokesman for the park confirmed his identity. His wife, who was also hiking in Yosemite, was injured and remains in hospital.
The couple, who were found with climbing equipment, are thought to have been scouting the ascent from a trail when a "sheet" of granite around 40m by 20m (131ft by 65ft) fell from a height of 650ft.
Scott Gediman, from the park, said: "With all the craziness I don't exactly know where they were going but chances are they were going up.
"From what I understand they were buried under rock... they were crushed by falling rocks."
Both suffered injuries "consistent with tonnes of granite falling on you," he added.
They were the only known casualties, despite being part of a group of other climbers when the collapse took place.
Witness Mike Kane tweeted that the collapse was "very loud", adding it was a "miracle more (were) not hurt".
On Thursday, a second rock fall at the site saw one person injured. They were airlifted to hospital.
Mr Foster's death is the first fatality from a rock fall in the park in 18 years, according to the parks service.
A Foreign Office spokesman said: "We are in contact with the local authorities and providing assistance to both families at this very difficult time."