First, it must be said that historians do not generally believe that anything is inevitable. Instead they usually see contigency, especially in complex events like the Cuban Missile Crisis that were basically created entirely by human agency. This being said, there is no doubt that American hostility toward Fidel Castro pushed him toward the Soviet Union in search of protection. The Eisenhower and Kennedy administrations had bitterly denounced Castro, and had attempted to spark a counter-revolution with the disastrous Bay of Pigs invasion in 1961. Additionally, the CIA planned several assassination attempts and otherwise actively sought to undermine his rule.
It did not necessarily follow, however, that Castro had to ask for nuclear missiles from the Soviet Union, which was the cause of the crisis. In addition, Khrushchev did not have to agree to the deployment, and indeed there was also plenty of disagreement within the Soviet Union on the wisdom of sending the missiles to Cuba. So while American policy toward Cuba drove Castro into the arms of the USSR, a move that he was ideologically inclined to make anyway, and certainly exacerbated Cold War tensions, it did not make a nuclear showdown of the scale of the Cuban Missile Crisis inevitable. It was created by human agency.