There is a notion that superpowers have, well, super powers, and can bend history to their will. There is never a shortage of conspiracy theories involving foreign agents. The reality of course is that even great powers are constrained, and the idea that Carter's appearance in Iran sparked a revolution is at the very least too Carlylian for my taste.
If Carter had a role to play, I would say that his administration placed too much faith in the shah's government, then was indecisive in its support. Carter managed to disappoint reformers even as he failed to provide unambiguous support for the-devil-we-know, and left the regime to its devices in the critical year of 1978 while his administration focused on SALT II and the Camp David Accords. Ultimately, forces in Iran, not a few provoked by the shah himself, are responsible for what followed.
The OP refers to New Year's Eve 1977. On a 16-hour visit, Carter became the first and only U.S. president to visit Iran, and at dinner made an infamous toast:
Iran, because of the great leadership of the Shah, is an island of stability in one of the more troubled areas of the world. This is a great tribute to you, Your Majesty, and to your leadership and to the respect and the admiration and love which your people give to you. The transformation that has taken place in this nation is indeed remarkable under your leadership.*
With that, Carter seemed to dash hopes that his administration would put real pressure on the shah, while at the same praising the very policies which had angered much of the Iranian public— inviting the scorn of the liberal, Marxist, and Islamist opposition alike. But to claim that this was the spark of the revolution is an overstatement.
In the early 20th century, as Persia was caught in the endless machinations of the British and Russian empires as they jockeyed for position, the U.S. may have been seen as a potential third power which would put a stop to the meddling. But the U.S. had no interest in Iran (it was cultivating influence with the Saud family on the other side of the Gulf) until after World War II, when Cold War politics came to dominate foreign policy. It was vital to keep the largest and wealthiest state in the Middle East out of the sphere of the Soviet Union. American and British involvement in the 1953 coup d'etat against the Mossadegh government was widely suspected, and later confirmed to be true.
Mohammed Reza Pahlavi by all accounts proved to be soundly anti-communist and pro-Western ally. To the West, he was abolishing feudal institutions, industrializing the economy, and secularizing Iranian society. The U.S. even curtailed intelligence-gathering in Iran in the 1970s, trusting his government to supply it. Internally, however, the shah managed to alienate large segments of the population with increasingly arbitrary rule (e.g. replacing the calendar): first clerics and landlords in the White Revolution of 1963, then the intelligentsia, and after he tried to blame the souring economic conditions of the 1970s on merchants, both the commercial classes and the unemployed.
The Carter administration did admonish the shah on human rights concerns early on, resulting in the token release of some political prisoners, but public protests began months before Carter's visit. Mustafa Khomeini, son of Ruhollah Khomeini, was killed while in exile in Iraq in 1977. The shah's secret police were widely believed to be responsible, leading to protests and a harsh crackdown. Secular intellectuals issued a challenge in October 1977, and citing this letter, Khomeini encouraged clerics to step up their opposition citing the letter. So the protests of January and February 1978 were the culmination of years of resentment against the regime, its secret police, and its propaganda machine, not a spontaneous critique of American foreign policy.
Some do blame Carter for later events. He had been elected partly on the promise of a government that practiced its high ideals. In contrast to the realpolitik that led previous administrations to turn a blind eye to human rights issues and social justice, his would not intervene in the affairs of other countries, and would take an active interest in disarmanent and peace. At the same time, however, Carter had the reality of serving a superpower's interests during the Cold War, in a period when the West had seen a succession of humiliations in Southeast Asia and Africa, and the U.S. was reeling from Watergate, Vietnam, and stagflation.
These competing interests played out at the very top of the administration. From the early months of the crisis, Zbigniew Brzezinski, the National Security Advisor, advocated for ardent support of the shah as in the best interests of the U.S. The State Department under Cyrus Vance (and the U.S. ambassador, William Sullivan) believed that the shah was a lost cause, and that supporting democratization would serve U.S. interests better.
By February 1979 Iran was in a state of civil war, but the U.S. dithered. Brzezinski was assuring the shah of unwavering support, the State Department was reaching out to the liberal opposition to try to smooth transition to a democratic government, but the government refused to support a coup d'etat or other action that could have replaced the shah with someone other than Khomeini, much less any direct military intervention. When the shah was admitted to the U.S. for medical treatment in October, students stormed the embassy and published documents showing that U.S. officials had met with moderate leaders. This helped turn public opinion against the moderates and raised fears of direct U.S. intervention, which Khomeini extracted for maximum propaganda value for years to come.