WASHINGTON—House Intelligence Committee chairman Devin Nunes is stepping aside from the panel’s probe of possible Russian interference in the 2016 election, citing the need to confront a congressional ethics inquiry into allegations that he improperly disclosed classified information to the public.

Mr. Nunes’s handling of the Russia probe has been criticized in recent weeks by Democrats who accuse him of providing political cover to the Trump administration rather than leading a serious, independent inquiry. The committee’s...