At the height of the teapot tapes scandal last week, Laws used his Radio Live morning show to blast the Herald on Sunday for its role.
He said: "If I had a gun I'd shoot them - put them out of their misery - because they have gone rabid and they may infect others."
That was followed by a segment questioning why listeners had not already carried out the shooting.
"Have you noticed ... the Herald on Sunday for example, which is rabid all the time ... no idea why somebody hasn't taken the shotgun there and just cleaned out the entire newsroom."
Laws' bosses will not say whether Laws' absence this week is the result of disciplinary action.
Mediaworks' radio manager Jana Rangooni would not explain why Laws was not filling his usual Radio Live slot, saying she could not comment on employment matters.
She would not say whether the station was backing his statements.
The Herald on Sunday said today would not lay complaints with the police or the Broadcasting Standards Authority over Laws' statements.
Editor Bryce Johns said the paper had a strong case for an official complaint but decided any action would "give oxygen to a fool".
However, he had spoken to "spooked" staff who were considering laying their own complaints.
"In the end he's just a man who is struggling for relevancy."
Police have not received any criminal complaints over the comments, a spokesman said.
Laws did not return calls.
