JERUSALEM—Israel’s peace camp and its international backers have long used one crude but powerful argument: Arabs make more babies than Jews and unless a separate Palestinian state is created, a demographic time bomb will turn Jews into a dwindling minority akin to white South Africans.

That prospect certainly seemed real when the Oslo peace process began in the 1990s. Fertility among Israeli Jews stood at an average of 2.6 children per woman, compared with 4.7 among Muslims in Israel and East Jerusalem and 6.0 among Palestinians...