BUDAPEST — David Pressman, a gay human rights lawyer, knew he was in for a rough time even before he arrived in Hungary with his husband and two children to take up a new job in September as the United States’ ambassador to Europe’s self-declared citadel of traditional Christian values and friend of the Kremlin. As his confirmation hearing began in July in Washington, a rubber dinghy carrying a warning appeared on the Danube River near the U.S. Embassy in Budapest. On a black banner emblazoned with a skull and crossbones was an anti-L.G.B.T.Q. message in English and Hungarian: “Mr....