The 'Skyfall' star was shopping for groceries with Weisz near their home when the fan tried to photograph them, reported Sun online.
During Thursday's rebuff, Craig reportedly screamed, "Is watching me food shopping with my wife really all that interesting to you?"
He then snatched the mobile off the shocked fan as Weisz tried to calm him down.
"Daniel was really angry, and it looked at one point like he might even destroy the phone," a source said.
The couple are known for being very private. Craig has previously said, "I think there's a lot to be said for keeping your own counsel. It's not about being afraid to be public with your emotions or about who you are and what you stand for.
"But if you sell it off it's gone. You can't buy it back. You can't buy your privacy back.