Even if any of these appointments were subject to confirmation, it would be the Senate that does the examining and not the House.
He wasn’t calling for confirmation proceedings, in the House or the Senate. He’s calling for a hold on any more appointments while backgrounds and qualifications of those already placed are examined. He didn’t say what form it would take. There’s no law prohibiting members of the House—or any one of us, for that matter—from looking into their writings, speeches, appearances, etc., just like Glenn Beck did. It’s what every member of congress ought to be doing.