The Navy's investigative board has recommended that the skipper of the USS Stark and the officer in charge of its weapons should be court-martialed for failing to take defensive action against the Iraqi warplane that fired two Exocet missiles into the ship the night of May 17, killing 37 sailors, authoritative sources disclosed yesterday.

The ship's executive officer, or second in command, Navy investigators said in a report still locked up in the Pentagon, should receive lesser punishment because he was not directly involved in the flawed decisions on how to respond to the approaching Mirage F1 fighter-bomber, according to Pentagon sources.

The hard-hitting report, based on a formal investigation conducted under Rear Adm. Grant A. Sharp, sets down a long list of alleged failures by Capt. Glenn R. Brindel, 43, of Pittsburgh, the Stark's skipper; and Lt. Basil E. Moncrief, 32, of Corpus Christi, Tex., its tactical action officer who runs the combat center where weapons are ordered into action.

The report cites everything from failure to warn away the Iraqi pilot by radio when he was not yet close enough to fire, to neglecting to prepare the chaff dispenser designed to confuse radar-guided missiles like the French-made Exocet, defense officials said.

Sources said the report does not recommend what kind of court-martial should be conducted, a decision to be made by higher authority. The seriousness of the court-martial could determine the severity of any penalties imposed if the officers were found guilty.

Lt. Cmdr. Raymond Gajan Jr., 35, of Rockville, the Stark's executive officer, is not in the same jeopardy because the board recommends nonjudicial action, such as a warning letter or reprimand, sources said.

The Navy announced June 19 that all three officers were being replaced in their posts on the Stark.

Defense Secretary Caspar W. Weinberger, who was traveling in Asia yesterday, will decide whether to accept the board's recommendations and go ahead with a formal court-martial or halt the legal process.

If he approves a court-martial, it would be the first during the Reagan administration resulting from a military event that resulted in loss of life.

A retired admiral conducted a nonjudicial inquiry into the 1983 Marine barracks bombing in Beirut that killed 241 U. S. servicemen, but no court-martial followed. President Reagan took responsibility for the tragedy.

The Sharp report, sources said, documents every defensive action the Stark should have taken but did not after the Air Force AWACS (Airborne Warning and Control System) plane orbiting at the western edge of the Persian Gulf warned U.S. warships, including the Stark, that an Iraqi plane was headed their way.

Rather than manning battle stations, the Stark remained in Condition Three, a lower state of readiness. The report is said to charge that the ship failed to prepare antiaircraft missiles, medium-range guns and close-in Phalanx weapons to fire.

Brindel wrote the House Armed Services Committee last month that he had told Moncrief "to keep a close eye on the contact" and recommended that the Navy determine why his subordinate had not done so.

The House committee's report on the incident also said Brindel had delayed a radio warning to the approaching plane.

Navy officers with experience in the Persian Gulf said Brindel and Moncrief had every right to think that the Iraqi plane was friendly and would swing away from the U.S. warship, as had happened many times. "Who in the Navy would have come to his defense if he had shot down a friendly?" one officer asked.

As in the Stark episode, the Navy in 1969 conducted a formal inquiry into why the USS Pueblo did not put up a fight before being captured by North Korean gunboats in 1968. The investigations board recommended that top officers of the surveillance ship be court-martialed. Navy Secretary John H. Chafee rejected the recommendation, stating on May 6, 1969: "They have suffered enough, and further punishment would not be justified."

The courts-martial, if they go forward, are likely to spread beyond the courtroom and intensify the congressional debate over the proper U.S. role in the Persian Gulf region.