Nearly four years after the mysterius death of a former top American diplomat's daughter, a criminal investigation is slowly dragging on, stalled by international uncertainties and public bickering between the State Department and a Virginia prosecutor.

Alexandra (Sasha) Bruce, the beautiful, meedy and rebellious 29-year-old daughter of the late ambassador David K.E. Bruce, was struck by a single gunshot on the right side of her head on November 7, 1975, at her family's Staunton Hill estate in southside Virginia. She died at a Lynchburg hospital 36 hours later.

Initially, her death was ruled a suicide. But in September 1978, a Charlotte County grand jury indicated her Greek husband, Marios Michaelides,, now 34, on murder, theft and other charges. But then, Michaelides had returned to Greece, where he has so far escaped prosecution. He has said he is innocent.

The latest snarl in the tangled case was brought to light in the past few days in an unusually sharp public exchange between the State Department and Charlotte County chief prosecutor Edwin B. Basker.

The State Department accused Baker of blocking a possible attempt by the Greek government to prosecute Michaelides by refusing to turn over evidence gathered during the Virginia investigation. "Anybody knows you've got to have a look at the evidence before you're going to prosecute," a State Department layer said yesterday.

I have had enough double talk from the State Department to last me a lifetime," Baker replied during a telephone interview yesturday. "I feel that I've gotten a runaround from them." Baker asserted that he would gladly rurn over his evidence to Greek authorities, but not to State Department officials.

The issue, the lawyers agree, is not whether Michalides should be extradited from Greece to stand trial in the United State. Such a move, they say, is barred under Greek law. Instead, the question is whether Michaelides will face prosecution in Greece on charges similar to those alleged in his Virginia indictment.

With the criminal proceedings at an apparent standstill, other Virginia officials stepped in quickly this month to try to speed up the case.

Gov. John N. Dalton held an unpublicized meeting July 16 with legal and administrative aides to discuss the case. Deputy Attorney General James E. Kulp, who was among those present, said yesturday that he would meet soon with Baker in an attempt "to get the thing off dead center." Baker also expressed hope that the state's intervention signaled progress.

John C. Lowe, Michaelides' Charlottesville lawyer, could not be reached for comment yesterday. Members of the Bruce family were also reported to be unavailable for comment.

Before his death in December 1977, Bruce -- who had been the top U.S. diplomat in France, West Germany, Britain and China -- indicated his suspicions about his daughter's death by initiating his own probe. Downey Rice, a Washington lawyer and investigator, was hired by the Bruce family to seek further evidence. Rice died of a heart attack in December 1978, shortly after the indictment was returned.

Baker and other Virginia officials said yesterday that the Bruce family has not contacted them recently about the investigation.

Alexandra Bruce, an honors graduate of radcliffe Collage, married Michaelides in a civil ceremony only three months before her death. Michaelides was previously married to a Tennessee woman Mary Lewis. One of the charges in the indictment is bigamy. Michaelides has denied the allegation, saying that his first marriage was legally dissolved.

Several suspicious details were that reportedly turned up during the criminal probe. Among them were that no spent cartridge was found at the site of Alexandra Bruce's fatal shooting and that the shot was fired from a gun belonging to Michaelides. Telephone calls were also reportedly made from the Staunton Hill estate to Tennessee, where Michaelides' first wife lived.

Baker and the State Department appeared prepared yesturday to let Greek authorities decide what to do about the investigation. "The State Department is not trying to urge any particular decision," said a department lawyer, who asked not to be identified.

I'm certainly willing to let (the Greek authorities) decide," said Baker