Obviously, anyone can bring a lawsuit at any time on any theory, but such as lawsuit would be dismissed early on in any of the forums in which it could be brought. It does not state a valid legal claim that can prevail in some tribunal that can enforce it.
What Is International Law?
International law is not a settled, consensus thing in the same way that domestic law is, so it is often not possible to say that something did or didn't violate international law or that a specific remedy is authorized. Even when it applies, it often only does so with the consent of the parties against whom one seeks to enforce it. One can develop a narrative based upon international law principles that argues one way or the other, but in the kind of fact pattern described in this question there is no authoritative answer.
There are doctrines and treaties out there that are often given effect in the domestic courts of various countries and in international tribunals which mostly rely on the governments of some country to enforce their rulings.
Analysis Of An International Forum
There are a handful of international tribunals that address very specific subjects like boundary disputes, trade disagreements in the context of an outstanding treaty, or war crimes arising under international law, but none of the existing forums would be appropriate for this dispute.
Only the U.N. Security Council claims jurisdiction to take action against sovereign states for generalized violations of international law and the U.S. would have veto power of any action it sought to take against the U.S. as a permanent member of the U.N. Security Council.
Analysis Of A U.S. Forum
A U.S. court would find that the doctrine of sovereign immunity applies and also might bar a case based upon the doctrine of statute of limitations, and would refuse to grant relief.
As a general rule, sovereign states claim immunity for all military and quasi-military and intentional actions they engaged in and waive immunity only their commercial activities and accidental and unintended harm to third parties (e.g. compensation for innocents harmed in war on the state's own terms even when the harm is justified by the laws of war).
Analysis Under Universal Jurisdiction Principles
Most world courts would find that the issue is non-justiciable or beyond their jurisdiction as this claim probably does not fall within the scope of universal jurisdiction which is the concept that any court with that jurisdiction may consider any matter that has worldwide impact (mostly war crimes or comparable severe human rights violations), which a CIA sponsored coup may have led to, but which were ultimately committed by the new regime and not the CIA itself.
Generally, universal jurisdiction is asserted in criminal or quasi-criminal proceedings against culpable individuals involved that involve harm to particular individuals, rather than against countries, so this is of limited relevance. The CIA agents involved in this event 64 years ago are likely dead or elderly and beyond the jurisdiction of any non-U.S., if they could be identified at all. (This is probably an important reason that CIA involvement was admitted at all.)
The Shah (who was forced into exile in Egypt with his family for the rest of his life) and many his leading cronies paid the price for their own wrong doing shortly after the 1979 revolution.
After the revolution, human rights groups estimated the number of
casualties suffered by protesters and prisoners of the new system to
be several thousand. The first to be executed were members of the old
system – senior generals, followed by over 200 senior civilian
officials, as punishment and to eliminate the danger of a coup d’État.
Brief trials lacking defense attorneys, juries, transparency or
opportunity for the accused to defend themselves, were held by
revolutionary judges such as Sadegh Khalkhali, the Sharia judge. By
January 1980 "at least 582 persons had been executed." Among those
executed was Amir Abbas Hoveida, former Prime Minister of
Between January 1980 and June 1981, when Bani-Sadr was impeached, at
least 900 executions took place, for everything from drug and sexual
offenses to "corruption on earth," from plotting counter-revolution
and spying for Israel to membership in opposition groups. In the 12
months following that Amnesty International documented 2,946
executions, with several thousand more killed in the next two years
according to the anti-government guerillas People's Mujahedin of Iran.
Analysis Of An Iranian Forum
The courts of the current regime in Iran might very well entertain such a suit as a violation of Iranian law, but would be hard pressed to enforce the claim. So, no criminal remedy would be possible given the unavailability of the defendants. Any civil or criminal money judgment or fine would in all probability be effectively impossible to satisfy short of an act of war against the United States.
In practice, any court judgment in this situation would be indistinguishable from a unilateral political act.
Also, even in the Iranian courts, entertaining a suit such as this one would require the current regime to claim continuity of regime with the one deposed in 1953. But, the political theory of the current Iranian state under the 1979 constitution (as amended), is that its authority rests upon the religious edicts of Islam and a referendum of the Iranian people, and not upon the 1953 constitution.
The Iranian Revolution of 1979 was a revolution against secular government generally and not merely the Shah. The preamble to the 1979 constitution specifically disavows both regimes as Wikipedia paraphrases:
The constitution begins by stating that the "anti-despotic movement
for constitutional government [1906-1911], and anti-colonialist
movement for the nationalization of petroleum" in 1950s failed because
of lack of religious coloring thereunder. Moreover, the "central axis"
of the theocracy shall be Quran and hadith.
Preamble further states: "The Assembly of Experts for
Constitution...fram[ed] the Constitution...[after input] by the
government...with the hope that this century will witness the
establishment of a universal holy government and the downfall of all
Also, the revolutionary doctrines of the 1979 revolution, while condemning the U.S. for supporting the Shah's regime, did not particular focus on the illegitimacy of the coup. So, there is no internal reason for Iranian courts to focus on this hook.
It is also plausible that treaties between the current regime of Iran and the U.S. would pre-empt this suit, although I do not have a comprehensive knowledge of those treaties. At a minimum, entering into multiple treaties could be construed as a waiver of any such claims, even if claims arising out of the 1953 coup were not expressly mentioned. The most recent treaty between the U.S. and Iran was entered into in 2015.