Secrets declassified can clarify hindsight. In particular, national security archives opened routinely just last year show why President Eisenhower took Sputnik so calmly, only to appear flustered at the American public's alarm. Ike, it appears, not only expected Sputnik, he welcomed it. What he hadn't expected was the overwhelming misperception that the United States was lagging behind in rocket technology.
The US and USSR promoted satellite launches as part of a peaceful research program called the International Geophysical Year (IGY). That was true enough, as far as public statements went. But not even top US civilian IGY planners knew Ike's real plan. He wanted to establish a new principle of "freedom of space" as international law. If national sovereignty stopped at the lowest satellite orbit, spy satellites could circle Earth just as warships freely roam the seas.
Wernher von Braun and his German rocket team working for the US Army could have orbited a satellite in January 1957. Eisenhower forbade this partly because he didn't want the appearance of a preemptive propaganda move to spoil the illusion of cooperation in space. When Russia hailed Sputnik as opening up the peaceful use of space, it unwittingly gave Eisenhower the freedom he wanted. The age of continuous surveillance by spy satellites was born.
Ike achieved his strategic goal at the cost of his political credibility. The public didn't understand why he wasn't "doing something" about this new "Soviet threat." He couldn't give the game away. Instead, he instituted the office of presidential science adviser and created the National Aeronautics and Space Administration.
America never had a "missile gap" with the Soviet Union. But without the public alarm, it's doubtful manned space flight and planetary exploration would have had the support to progress as far as they have today.