Self-inflicted damage following the destruction of Iraq's nuclear reactor has personified Prime Minister Menachem Begin as a potential threat to Israel's indispensable relationship with the Reagan administration, arousing anguish among Israel's strongest U.S. backers.

An additional irony for Begin is that astute diplomacy be U.N. Ambassador Jeane Kirkpatrick has now put the United States and Iraq in tandem, a defeat for Begin's strategy of keeping the United States and Arab world split.

The anguish about Begin was typified by an unlikely source when Se. Paul Tsongas (D-Mass.) told State Department officials testifying before the Senate Foreign Relations Committee last week that "a distinction" much now be made between Begin and Israel. Tsongas has a 100 percent pro-Israel voting record.

The anti-Begin backfire is raising for Israel the specter of partial separation from the United States, its only benefactor. Even before the prime minister's perfectly executed order to destroy Baghdad's nuclear facility, President Reagan privately expressed contempt for Begin's attack on West German Chancellor Helmut Schmidt last April. The invariable polite Ronald Reagan found Begin's language unbecoming a head of state.

Far more harmful are Begin's unproven justifications for the raid, undermining Israel's credibility in the Reagan administration. His government's formal explanation of the raid asserted that Iraqi President Saddam Hussein had said the reactor "was being built against Israel alone." A full retraction had to be made. The formal government statement said that "no foreign expert was hurt" in the raid, another provable misstatement.

Begin then zeroed in on Secretary of Defense Caspar Weinberger, one of Reagan's closest friends in the administration. Within hours of Weinberger's persuasive argument at the secret June 10 National Security Council meeting for a strong American response to the raid, Begin had been informed. The spectacle of a foreign leader attacking one of Reagan's Cabinet chiefs for advising the president at an NSC meeting did not set well in Reagan's White House.

Behind the scenes, Ambassador Kirkpatrick was pulling off her diplomatic feat: persuading Iraq to tone down its Security Council resolution to a level acceptable to the United States. The unwitting catalyst of that achievement for both the United States and Iraq (which has long desired closer relations with Washington) was Menachem Begin himself.

Mrs. Kirkpatrick could not have pulled her coup without preliminary groundwork by two of the closest friends the United States has in the Arab world: Saudi Arabia and Egypt. President Anwar Sadat, cut out of the Arab community ever since the Camp David accords, privately encouraged the Saudis immediately after the Israeli raid to send this message to Iraq's Saddam Hussein: go easy on the United ystates and Ronald Reagan. They argued that Israel's offense should be exploited to tighten, not weaken, U.S.-Arab links. The Saudis had similar hopes.

When the Arab foeign ministers met in Baghdad following the raid, Saddam Hussein was impressed by the arguments of Saudi Foeign Minister Prince Saud. Saud warned that in the aftermath of the raid Begin would use the expected harsh anti-U.S. Iraqi reaction to further weaken Iraq's standing in this country. Don't let that happen, Saud argued; exploit Begin's exposed position to stregthen the U.S.-Iraqi link.

Other Arab leaders sang the same song. If Iraq accepted the veiled Israeli implications tying the United States into the raid, and blamed the Reagan administration, the old familiar cycle would be repeated, with U.S. television networks and the press gorging on Arab infamy. Saddam Hussein bought that package. His June 8 statement attacking Iran and "the Zionist entity" for the raid ignored the fact the U.S.-SUPPLIED f16s were used to destroy his nuclear reactor.

It was Ambassador Kirkpatrick's unenviable assignment to sit for long hours with the Iraqi foreign minister and hammer out the compromise. Behind the scenes, she had sympathy and support from some leaders of the American-Jewish community who know that long-term U.S. relations in the Arab world can make or break President Reagan's attempt to build an American-Arab regional defense system for the Persian Gulf.

In succeeding, Mrs. Kirkpatrick not only preserved but actually strengthened those relations. Her success notified Prime Minister Begin that Reagan's fidelity to Israel cannot be used to subvert the infinitely higher consideration of America's own national security.