Israel's the one that started that war, though to be fair, it was pretty much a foregone conclusion that other middle eastern countries' actions would lead to that result. The more important factor is what Israel itself did to get said other countries to do those actions. Wikipedia has a summary of said actions here:
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Summary of events leading to war
After the 1956
Suez Crisis, Egypt agreed to the stationing of a
United Nations Emergency Force (UNEF) in the Sinai to ensure all parties would comply with the
1949 Armistice Agreements. Despite the overwhelming support for Resolution 1000 in the UN General Assembly (UNGA), Israel refused to allow UNEF forces onto its territory. In the following years, there were numerous minor border clashes between Israel and its Arab neighbors, particularly Syria. In early November, 1966, Syria signed a
mutual defense agreement with Egypt. On November 13, 1966, in response to PLO (Palestine Liberation Organization) guerrilla activity, including a mine attack that left three dead, the Israeli Defence Force (IDF)
attacked the village of
as-Samu in the Jordanian-occupied West Bank. Jordanian units that engaged the Israelis were quickly beaten back. King
Hussein of Jordan criticized Egyptian President
Gamal Abdel Nasser for failing to come to Jordan's aid, and "hiding behind UNEF skirts". Israel was censured for this invasion in
United Nations Security Council Resolution 228, being reproached by the US, the UK, France and the USSR. On April 7, 1967, Israel invaded Syria. The USSR urged that the collective defense agreement with Egypt had been triggered. In May 1967, Nasser received false reports from the Soviet Union that Israel was massing on the Syrian border.
Nasser began massing his troops in the Sinai Peninsula on Israel's border (May 16), expelled the UNEF force from Gaza and Sinai (May 19) and took up UNEF positions at
Sharm el-Sheikh, overlooking the
Straits of Tiran. Israel reiterated declarations made in 1957 that any
closure of the Straits would be considered an act of war, or justification for war. Nasser declared the Straits closed to Israeli shipping on May 22–23. On May 30, Jordan and Egypt signed a defense pact. The following day, at Jordan's invitation, the Iraqi army began deploying troops and armored units in Jordan. They were later reinforced by an Egyptian contingent. On June 1, Israel formed a National Unity Government by widening its cabinet, and on June 4 the decision was made to go to war. The next morning, Israel launched
Operation Focus, a large-scale surprise air strike that launched the Six-Day War.
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Source:
en.wikipedia.org