That is so not true. USA officially congratulated USSR, and the president (Kennedy) himself sent a personal message to the Soviets, lauding their achievement. Among common people, there was suspicion about USSR's real intentions (about space research aiding missile research), and disappointment that they were behind the USSR, but that was still healthy competition, not any beliefs in nazi ideologies of super race. And there was plenty of excitement about what the future held for space travel, which the event had proved was possible.
It also strengthened their resolve to put a man on the moon before the soviets, which they did. All in all, it was a spirit of comptition in science and technology that bnefited everybody - ironically, it was the soviet philosophy that disdained competition and entreprneurship in favor of governmental monotony. The space race itself was a triumph of americal ideals of competition bringing out the best in everybody.