Why would he? You seem to be operating under the assumption that enduring hits from Superman is a relevant feat to Godzilla. It isn't. Superman likely couldn't hurt Godzilla even with his best bullrush.
Godzilla's mid-range durability feats (tanking nukes and city level blasts) are comparable to the absolute highest end durability feat Superman has, and Supes was almost killed by the nuke he took. Given Superman's punches can't do serious harm to other beings with his durability (Zod, Doomsday, etc) there is no reason to think he could really hurt something significantly more durable than himself.
Forget Steppenwolf. Even Superman likely can't pose a serious threat to Godzilla.