Probably the best place to start is with the various coups d'etat in Turkey.

In 1960, a military group overthrew the democratically-elected government in Turkey. Many of the soldiers had previously been trained by the U.S. as an anti-communist stay-behind group. They immediately pledged allegiance to NATO. They established a coalition government. In 1971, the Turkish military again took power, this time through a memorandum. In 1980, the military again took power until 1982. In 1997, the military forced the prime minister to sign a few referendums and resign (referred to as the post-modern coup). In no instance did the U.S. suspend its relationship with Turkey.

Moving across the Aegean, in 1967, the Greek military overthrew its government. The U.S. initially suspended heavy military equipment, but resumed it in 1968 after the Soviets invaded Czechoslovakia. Then the U.S. suspended it again until 1970. In 1971, Congress passed an act that would force the Nixon Administration to suspend all military assistance to Greece until Democracy was restored, but the President kept assistance going through waivers.

In 1974 in Portugal, the corporate statist (related to Mussolini's fascism) regime was overthrown in a bloodless coup that linked up with the carnation revolution. The U.S. was initially distrustful of the new government, but in the end decided to support it with an aid package.

In Spain, in 1981, there was a coup that lasted for 18 hours (known as 23-F). As the coup did not ultimately succeed, I won't go into it too much more, but it is notable not for any international reaction, but for the role of the monarchy in ending the coup.

As you can see, NATO has never attempted to intervene militarily in any of these instances. The U.S. did cut off foreign aid in the case of Greece. It has been U.S. law since 1961 to suspend aid to any country that overthrew a democratically-elected government through military means until democracy is restored. (In the current wording, the Secretary of State has to certify to Congress that democracy has been restored.) The U.S. has been inconsistent with applying this throughout its history.