The Arab-Israeli conflict could be seen as a sideshow to the proxy conflicts that were common in the middle east.

Initially both the US and the USSR had limited influence in the regions as they had been the colonial territories of European empires. However many young Arabs had been heavily influenced by Marxist anti-imperialist ideas, leading to left wing governments such as Nasser in Egypt. These new states reached out for foreign backing, both to the USSR and the West, ultimately choosing assistance from the USSR.

This was usually because the USSR gave them military and economic support cheaply (or even for free) in the hope for expanding the revolution and gaining influence in the region.

During this time the West lacked few friends in the region so they started supporting this western originated country (insert complicated population makeup/racism here) militarily to keep influence in the region.

The actual conflict itself have limited role in the wider Cold War, though the 1973 Yom Kippur war did spark a short diplomatic standoff. Really the middle east proved a failure for the USSR as most of its clients apart from Syria moved away from it during the 1970/80s.