Former National Security Agency contractor Edward Snowden has revealed so much information about government spying in the past two years that little seems shocking. But allegations in his latest leak, published by the Intercept, could upend any chance the White House has of mending relations with Silicon Valley in the near future.

The Intercept, citing information from Mr. Snowden, alleged that the NSA and British spies hacked into the computer network of Gemalto NV, a Dutch company and the world’s largest manufacturer of...