WASHINGTON—The U.S. struck and destroyed three coastal radar sites in Yemen early Thursday in a significant military response to a series of attempted attacks against American warships in recent days, deepening America’s role in the country’s protracted civil war.

U.S. personnel aboard the destroyer Nitze launched a series of Tomahawk cruise missiles against the three separate radar sites along the Red Sea Coast, north of what is known as the Bab-el-Mandeb strait, Pentagon officials said in a statement late Wednesday. Initial...