The Senate approved legislation Friday that would protect the privacy of video store customers, an outgrowth of newspaper accounts of former Supreme Court nominee Robert Bork's viewing habits.

The legislation, passed by unanimous consent, would create a federal law to protect the rental records of video store customers. Such stores would be required to obtain the consent of customers before lists of the movies they rented could be made public.The legislation has not yet been acted on in the House.

The video privacy act was proposed by Sens. Paul Simon, D-Ill.; Patrick Leahy, D-Vt.; Charles Grassley, R-Iowa; and Alan Simpson, R-Wyo. The four are members of the Senate Judiciary Committee, which held hearings last year on President Reagan's nomination of Bork to be a member of the Supreme Court.

During those hearings, a weekly newspaper in Washington, D.C., published a list of movies that Bork and his family had rented from a local video store, angering Bork and members of the committee, who felt it was a violation of privacy.