Keep in mind a couple major caveats:

The US was demographically, economically, and militarily superior to all its allies, making it a default leader in its bloc. In contrast, China had a larger population than the Soviet Union and so it could contend on which nation was more significant to the communist bloc.

Also, Mao had seniority over communist leaders in the Soviet Union after Stalin died - and without democratic elections for leaders in the communist bloc, there wasn't a natural rotation of leadership that would prevent any one leader from claiming to be the senior guiding figure.

The bloc aligned with the US wasn't always capitalist or democratic - so there wasn't near as strong of a drive for 'leadership' over an ideology as much as unity against communism. In contrast, the various leaders of the Soviet Union and China had differing ideological bends: there was Stalinism, Marxist-Leninism, Maoism, etc.

The closest you'll probably get is when DeGaulle pulled France from NATO's military command over varying disagreements, including the fact the US wanted European nations to pay for its troops stationed in Europe, lack of a guarantee of protection from nuclear attack, and a push for French sovereignty in world affairs, especially after the US turned against the British, French, and Israelis during the Suez Crisis.

Of course, a lot of this was also for show, as France had secret agreements that France would walk straight back into NATO's integrated command structure if hostilities broke out:

De Gaulle's defiant gesture, which caught Washington unaware, came at a time when U.S.-European security revolved around girding against a possible Soviet attack from the East. It meant in theory that the French military and its nuclear arsenal would no longer take orders from the American general commanding NATO forces. In addition, de Gaulle ordered out thousands of U.S. troops stationed on French soil and at NATO headquarters, then in a Paris suburb.

France never left the overarching North Atlantic Alliance, however, and within a year the practical effect of withdrawing from the integrated command was also watered down. A secret accord between U.S. and French officials, the Lemnitzer-Aillert Agreements, laid out in great detail how French forces would dovetail back into NATO's command structure should East-West hostilities break out.

u/technique_only's comment was good I just want to add to it. It's important to remember that while stalin was alive the communists really did function as fairly monolithic bloc directed from moscow. Stalin exercised far more control over his fellow countries than any American president exercised over the any NATO countries, to say nothing of communist movements around the world. When Stalin died, Kruschev expected to continue with that level of control (and did do so with eastern europe), while Mao seems to have expected some of it to fall to him, or at least parity within the soviet bloc. the tension that ensued that led to the split was only really possible because the degree of control the USSR had over his ideological allies was so much greater than any the US ever wielded.