WASHINGTON—Turkey moved closer to severing its relations with Israel, demanding that Prime Minister Benjamin Netanyahu publicly apologize for his government's high-seas military action against a pro-Palestinian flotilla this week in order to avert a formal diplomatic rupture. Senior Israeli officials responded Friday that their government would never apologize for an act of "self defense" and acknowledged that Israel could be on the verge of losing its closest military and economic ally in the Middle East. Such a development, these officials said, would raise new strategic and diplomatic challenges for Israel if Ankara reorients itself away from its historically pro-Israel and...