I am currently finishing a Master's Thesis on US-Czechoslovak relations during the Prague Spring, and so, while I do not quite have my Master's degree, I can certainly answer this question with the required expertise. Any good description of the "official" response of Lyndon Johnson's administration, however, should be juxtaposed by the administration's unofficial response, as well as the ways in which the administration used the invasion to further it's foreign policy agenda.

The official response: Most notably, the Johnson administration cancelled a proposed summit meeting to be held between Johnson and Alexei Kosygin, the Soviet premier. The United States also protested the invasion at the UN, interestingly against the wishes of the Czechoslovak foreign ministry. The Johnson administration also warned the Soviet Union about possible American action if the Soviet Union invaded Romania, which, under the leadership of Nicolae Ceaușescu, had in reality shirked Soviet international initiatives in a way that Dubcek had never considered doing (Dubcek's staunchly pro-Soviet values are another discussion altogether). The Johnson administration, however, took no real aggressive action against the Soviet Union, and its lackluster response has rightly been remembered by historians of American foreign policy as such (note that Czech scholars of foreign policy today criticize Johnson for his "reserved" response, especially given that the United States was supposedly at war with international Communism in Vietnam at the time, though I have to admit I rely here on a Czech college textbook Ceskoslovenska a ceska zahranicni politika: minulost a soucasnost (Czechoslovak and Czech Foreign Politics: Past and Present), and if you do get your hands on this book, see pages 308-310).

The unofficial response: What's much more interesting in this case--and this explains Johnson's lackluster official response--is the unofficial, clandestine response of the Johnson administration to the invasion. To understand this aspect of the administration's response, you have to understand Johnson's primary foreign policy goals in the year 1968, the same year as the Tet Offensive. Johnson wanted to end the Vietnam War and to bring about detente (a relaxation of tensions) with the Soviet Union. He had announced in March that he would not seek reelection, and he meant to succeed in improving American global interests after the disaster in Vietnam. Johnson and his influential Secretary of State Dean Rusk worked very hard to organize the eventually cancelled summit meeting with Kosygin which would be aimed at reaching a significant arms reduction deal with the Soviet Union (in his memoir, Rusk argued that arms reduction talks "were of transcendent importance," standing above all other foreign policy considerations, specifically the fallout from the invasion of Czechoslovakia; see pg. 350 of his "How I Saw It."). This consideration, combined with a widely held American perception that Czechoslovakia rightly existed within "Eastern Europe" and the "Soviet Sphere" (the history of this perception is another conversation as well), led to a response by Lyndon Johnson that might be hard for some to fully accept: he entirely dismissed the Soviet Union's official declaration of the invasion, delivered by ambassador Anatoly Dobrynin, in favor of continuing discussion about the planned announcement of the Summit meeting, which was (the announcement, that is) scheduled for the morning (American time) of August 21st -- the morning after the invasion. As is recorded in Johnson's White House tapes, Johnson told Dobrynin that he had already explained to the ambassador his views of the "crisis" in Czechoslovakia--indicating both that he perceived a crisis in Czechoslovakia, and that this crisis was not the invasion! (here is the official State Department transcript of that conversation, based on the tapes). Dobrynin said in his memoirs that he couldn't believe that the "utterly oblivious" Johnson still thought they could hold the summit meeting. After meeting with his National Security Council, Johnson realized that public opinion, both domestic and foreign, would prevent him from holding the meeting and he did agree to cancel it (i.e., the official response). He, however, almost immediately began negotiating behind the scenes, both through Dobrynin and the American ambassador in Moscow, for a rescheduling of the cancelled meeting. Thus Johnson's unofficial response was to entirely ignore the invasion in favor of detente, which explains his lackluster official response.

How the administration used the invasion to its advantage: The administration did use the invasion to shore up support for NATO (which was scheduled to be terminated) and to keep troops in Germany (many in Congress wished to reduce defense spending at this time). This is well covered in popular accounts of the Cold War, such as John Gaddis's The Cold War: A New History, so I won't worry about going into detail here.

In the end, the American response to the Prague Spring is remembered as a sort of tragedy for Johnson, who failed to initiate detente with the Soviet Union during his presidency (that honor went to his successor Richard Nixon). What popular accounts of Johnson's presidency and the Cold War have so far ignored, however, is just how little Johnson cared about Czechoslovakia, and how much he was willing to overlook in order to pursue detente.

Most of this analysis is based on the documents found in the Foreign Relations if the United States series, and you can find the list of pertinent documents here..

I don't know enough to answer your question about the reactions of European Nations, other than to say that the invasion obviously scared them enough to rally support for the continuation of NATO.