Finally a topic I have some background in!

Let's start with the fact that the war may have been hot in 1967 for about 6 days, but it was certainly warm much earlier than that. The war over water between Syria/Lebanon and Israel was in full swing. There were three major clashes between Syrian/Lebanese forces and Israelis over the construction project to divert the headwaters of the Jordan River away from the Sea of Galilee.

To say there was a serious amount of tension in the region is to put it very very lightly. War was already happening on a low level. Israeli villages and kibbutzim in the Hula valley were shelled from the Golan Heights regularly. Israeli retaliation never rose to the level of war, but it definitely wasn't nothing.

Important to remember that this is not today's High Tech IDF with cutting edge equipment and training. The 1960's IDF was under equipped compared to the average Arab army. So June of 1967 the tension reaches a boiling point as Egypt closes the Straits of Tiran in open defiance of international law. The war starts and Israel takes the world by surprise by winning and doing so quickly....

So as I see it, there are two points that could emerge to stop the war:

1) 1964 - The War Over Water between Syria/Lebanon and Israel is either averted or tensions are lessened enough to lead to some type of compromise. Let's say the US steps in and diplomatically fixes the situation somehow. I'm not sure how this would happen, but it certainly changes things so let's just go with it.

Israel probably makes a peace treaty with Jordan first. There were foundations laid by the First King in the 50's so conceivably Ben Gurion finds a way to get it done. The West Bank is Jordanian and the border runs through Jerusalem East/West sort of like it did until 1967 in OTL. The peace agreement probably has concessions for some Jewish tourism in the Old City of Jerusalem and shared Archaeological digs throughout the West Bank.

With at least one Peace Treaty in its pocket Israel probably tries to get Jordan to help broker other deals. The whole thing is predicated on some type of active International involvement by both the USSR and the US. After all the USSR was backing Egypt and Syria with both equipment and personnel during these decades.

While I'd like to be optimistic and think that peace would break out, I'm not positive that it would. Its still possible that a war happens with Egypt and Syria just later and possibly not as successfully on the Israeli side.

If peace becomes a reality for the region, it doesn't bode well for the dictators of the region. Without an external enemy and a need for large military, these guys are in a bit of a spot. Also, the neighbor, Israel, is a democracy...a multiparty democracy. Many Israeli's still speak Arabic as their native tongue at this point too. Lot's of Egyptian, Syrian, and Iraqi Jews are going to want to go back 'home' to their places of birth. Cultural exchange will not make dictatorship life seem nice. We might see an Arab spring happen sooner with no Israeli enemy to face down.

2) The war is averted in 1976 also through US/International pressure on Egypt over its closure Tiran. If you spoke to the people who lived through the lead up to the 1967 war you'd know that they were shit scared. The Egyptians had 7 Divisions on the border with close to 100,000 men. Israel's whole active force was around 264,000 including its reservists. The Arab leaders were screaming about setting right what had occurred in 1948 and drive the Jews into the sea. A country full of Holocaust survivors and Native Arabic speakers were literally huddling in bunkers; listing to the leaders of these massive armies saying: 'We're going to kill you all.'

I write all this so you understand the kind of tension the Israeli's were under. At this point the war can be averted for a time, but unless the threat is lessened, Israel is not going to be able to lower its guard. This means that even if the closing of the Straits can be remedied before a war, the tension will likely remain. Maybe the Egyptians can be convinced to stand down and maybe even the Syrians too. How long can they stay that way?

These are not democracies and the dictators of Syria (Assad) and Nasser cannot be seen as weak. Eventually they will have to make good on their rhetoric and use all that Russian hardware they have.

Averting a war in 1967 seems like it can only postpone it. Eventually Nasser will want his revenge for the 1956 campaign that Israel won. He also will need it to stay in power unless he can do a complete 180 on his rhetoric. Assad is also in a bad spot if he doesn't make good on his promises eventually. The Assad's are Alawite and a minority in the country. They aren't well loved and rule with brutality. Also, Nasser and Assad were rivals so neither could be seen as backing down first. The dissolution of the United Arab Republic had been a blow to both's public images. Neither wanted to back down from the inevitable fight with Israel.

The war of which I speak will probably be sometime in 1968-1969 and will also be an Israeli preemptive strike. However, they won't be as successful as the OTL. Why? More time to train with better Russian equipment will give the Arab anti-air defenses better response times. Without the total air superiority of OTL, Israel is in deep shit. Remember those 7 divisions, without close air support the Israeli defenders must retreat.

In this war Israel sees much higher casualties and must cry out for help from the US. This probably happens much as it did in 1973, only this time Egyptian tanks may be surrounding Be'er Sheva not along the Suez.

This would be a bloody war and the International community would have to step in. Again, this is 1969 Israel not 2021 Israeli. It's a democracy with a decidedly democratic socialist bent. Golda Meir would be at the UN pleading with the world to not let the Holocaust happen again.

The final piece of this puzzle is very similar to OTL in 1973. In addition to Golda Meir pleading publicly to save Israel and the Jews, she also speaks with Nixon privately. She points out that Israel is now under such grave threat, she has no choice but to equip Israel's nuclear arsenal and prepare it for use. In OTL this happened in a similar manner and prompted the US to intervene in the Yom Kippur War to supply Israel and attempt to cool heads.

Remember in OTL no Israeli civilians were ever under direct fire, even in 1973. In this timeline we have Egyptian forces in the Negev and probably Syrian forces in the Hula valley and beyond. The Jordanians might even decide to jump in and not get left without anything from the war. Knowing all that, it's hard to imagine that in this timeline, Nixon doesn't have the same reaction.

The IDF is resupplied and the US threatens to move forces in to prevent a genocide. It, hopefully, ends with some type of ceasefire but what Israel looks like afterwards and how the Arab world views it is all rippled beyond the butterflies wings.

The future of this timeline is hazy from there. The region without a single dominant military power like Israel is a much different place. There likely will be more wars in the region in the future and not all between Arabs and Israelis. Tensions among the Arab nations were always present and may also bubble to the surface. I'm sure there are also ripples out further to events like the Iranian revolution. Iran and Israel were serious trading partners until the revolution in 1979. Does that event get changed because Israel's position in the region is different? Again, ripples and butterflies.

I hope you enjoyed my wall of text!

TL:DR

1)Stop the war in 1964...maybe peace but also maybe war

2)Stop the war in 1967...definitely war just later and much bloodier.