Well if he’s got proof that this actually happened then I’d say Obama would loose any support he has among Democrats in Congress.
It would depend on how the media plays the story—if they suppress it or persuade the low-information voters that it is no big deal, then the Democrats may not make a fuss. They would figure that any information gleaned would be used against Republicans. On the other hand, they might be worried about the White House being able to blackmail them so they would toe the Obama administration line.