When Eric Schmidt was the chief executive of Google in the mid-2000s, he dated Marcy Simon, a New York-based public relations executive. Both were married to other people at the time.

Mr. Schmidt and Ms. Simon were seen together on the French Riviera, at tech conferences and on Fire Island, where she owned a beach house. When a large yellow diamond ring was spotted on Ms. Simon’s finger, some speculated in the media that Mr. Schmidt might divorce his wife and marry Ms. Simon.

But Mr. Schmidt moved on to other girlfriends. Although he and Ms. Simon, who divorced her husband, later rekindled their relationship in the late 2000s and early 2010s, they decided to go their separate ways in 2014, people with knowledge of their relationship…