I don't remember the particular book, but I do remember reading that the Israelis communicated to Nixon that they were going down the tubes because of the quantities of first-class Soviet equipment that the Egyptians and Syrians had, and that if things didn't change in a day or two that they were going to nuke the Arabs - EVERYWHERE. Including the Aswan Dam and the Saudi oil fields. Now, this was a book and not tablets handed down from God, but I remember that the author was someone with some high degree of credibility. I don't know that the Israelis would have gone that far, but if you're looking at a second genocide within a generation it isn't too far-fetched to say "screw everyone else if they're doing this or letting it be done to us."
That’s probably true. Golda Meir is said to have communicated to Nixon over the phone that Israel wouldn’t be going quietly, even if it meant problems for the US. However I doubt that was the basis of Nixon’s decision. I think he viewed it as a proxy cold war battle, which he didn’t want to lose. And in his memoirs he’s indicated the greatest mistake of the Eisenhower administration was pressuring Israel, England and France to return the Sinai and Suez canal in anticipation of Egyptian gratitude. Which was returned by an Egyptian move toward the Soviets.
It has been well documented that Israel was on the verge of using nukes at the darkest moments of the Yom Kippur War.
With the Syrians advancing on the Golan and the Egypt’s successful push across the Suez Canal, the Israeli position was dire; the IDF had lost a significant number of aircraft due to the SA-6 missile system which had been supplied to Egypt, and armor units took heavy losses from anti-tank missiles.
There are two versions of the story; the first says that Defense Minister Moshe Dayan, fresh from a visit to the Golan front, was badly shaken by Israeli setbacks and urged Golda Meir to ready the nuclear option, but she refused.
http://www.timesofisrael.com/dayan-pushed-pm-meir-to-consider-using-nuclear-weapons-in-1973/#!
This version slightly contradicts earlier accounts, which suggested that PM Meir had ordered the assembly of some air-delivered weapons. At least one former U.S. official has also reported that Israeli Jericho missiles were placed on heightened alert; they were also capable of delivering a nuclear warhead.
The unsung heroes in all of this were the aircrews, maintainers and support personnel of the USAF’s Military Airlift Command (MAC) and the SAC tankers that provided in-flight refueling. After President Nixon gave the order, M-60s were pulled out of our armored battalions, loaded on C-5s and sent to Israel. F-4s were pulled out of Air Force squadrons at Seymour Johnson, Eglin and Nellis and flown to Lod Airport in Israel. They arrived with USAF markings and most flew in combat with the Star of David over the American camouflage paint scheme.
The airlift saved Israel, and made the nuclear option unnecessary. It also highlighted the value of the much-maligned C-5A Galaxy; no other airlifter could carry tanks and other out-sized cargo such long distances.