The US not only disapproved of the tripartite intervention in Suez, they very actively worked to oppose it. Their methods ranged from general diplomatic opprobrium, verbal reprimands from leading figures in the American government, most prominently Secretary of State John Foster Dulles and President Eisenhower himself, massive financial pressure, and even subtle but distinct military threats. In fact, during the Crisis a US Navy battle group with 5 aircraft carriers, including the USS Forrestal, the largest warship in existence then, was deployed to the Mediterranean with ambiguously threatening orders to 'prepare for imminent hostilities'. To put things into perspective, US warships of that tonnage in recent history were deployed in the First Taiwan Straits Crisis (1954) and the Dardanelles Crisis (1946), both very high stake, high tension situations. This gives you an idea of how seriously the Eisenhower administration opposed the Anglo-French scheme.
'Dictate' implies an overwhelmingly coercive force. Crucially, the reason why the lack of US support dictated a British withdrawal was because the Suez crisis unfolded DURING and indeed BECAUSE OF the British empire's declining power in global affairs relative to that of the US. The British invaded Egypt to regain control of the Suez Canal, which was essentially an incredibly strategic infrastructural asset operated by the British (and French, to some degree) but had been recently confiscated by the ultra-nationalist Nasser government of Egypt. The Canal was of immense importance to the British empire--it was the primary sea route between the home islands and key Imperial and Commonwealth territories in the far east e.g. Singapore, Hong Kong, Australia, etc. The British, particularly after the enormous costs of fighting two world wars, could not support a sufficiently large naval force capable of defending both national possessions in and around the UK as well as colonial holdings further in the east. This resulted in a situation where they had become a 'one hemisphere navy defending a two hemisphere empire'. The Suez Canal was critical to ensuring British naval forces could deploy swiftly around the globe to deal with threats. Huge amounts of essential commodities which the British economy depended on like oil also travelled through the Canal. Britain began redirecting cargo and trade around the Cape of Good Hope instead, at immense additional costs. When Nasser cut off maritime trade flowing through the Canal, the British suffered significantly, at some point even imposing the infamous Four Day working week and fuel rationing because key sectors of the economy were buckling under the dual strain of postwar austerity and Nasser's actions.
Enter the US. At the end of World War 2, Britain's total credit deficit was nearing 1.3 billion pounds, with the US being its largest creditor. The Crisis was worsening this by painful margins--by November, the Treasury reported a further loss of 279 million pounds due to a run allegedly instigated by American traders and bankers. As British imperial power receded, the Americans, backed by their massive industrial capacity, were also starting to become the predominant global superpower. The Americans blocked Britain's request for IMF emergency funds (which the British desperately needed to support the current value of the pound), and Eisenhower himself refused to waive legal restrictions preventing American oil companies from selling oil in bulk to the British. He also personally obstructed the Middle East Emergency Committee, an institution specifically designed to handle situations where the Canal was blocked to international trade, saying that the British had behaved irresponsibly and should be 'left to boil in their own oil, so to speak'. The British economy simply could not sustain this amount of pressure.
A further reason is that at that point in time the British had not developed sufficient weaponised nuclear capabilities and were hence relying on the American nuclear umbrella to guard against Soviet aggression. Negotiations were ongoing to deploy American Thor missiles in East Anglia; the 1962 Nassau Agreement gave American military officials permission to mount the Polaris nuclear missile system on Royal Navy submarines at Holy Loch, the catch being that the Americans, not the British, got the final say when it came to targeting and firing the missiles. Sure, the British had their own nukes, but their primary method of delivery was by strategic bombers which at that point were considered generally less effective and reliable than missile-based platforms, since the Soviets demonstrated significant ability to shoot down high flying jets quite accurarely (as demonstrated by the Gary Powers incident just a few years later).
In summary, a declining Britain was too dependent on a rising America in too many critical aspects of that time.
This is a topic I have great interest in, and I would be more than happy to elaborate further. I have left out many details as I feel they would take too long to explain and draw in too many related but distant issues!
I am currently away from home so I apologise for the sloppy citations, will provide more details when I can.
Some Sources: *Just Another Star? Anglo-American relations since 1945 *Money, Oil, and Empire in the Middle East: Sterling and Postwar Imperialism *A Special Relationship: Anglo-American relations from the Cold War to Iraq *The Suez Crisis
Actually it was so that as of July-August 1956, Eisenhower was supportive of an intervention in Egypt, and Dulles (his Secretary of State) openly encouraged it.
However, as the time passed (EDIT: the British and French militaries were 'busy abroad', i.e. involved in a number of wars overseas, had to withdraw some of involved units and re-train them, prepare necessary shipping etc., and thus took much too long to prepare an invasion of Egypt), the US administration became concerned that an invasion of Egypt short of US elections (in November 1956), would be inopportune. This was so because the US public was generally 'anti-colonialist' at the time.
Another issue was that, subsequently, London and Paris left Washington in darkness about their planning. They didn't announce their attack on Egypt.
Thus, when the British and the French then did launch their invasion, in late October/early November, just days short of US elections, Eisenhower was outraged and began exercising economic and military pressure for the operation to stop. Between others, he supported a UN resolution denouncing the attack on Egypt. Issued on 2 November, this called for an immediate ceasefire and was followed by US threats to cancel vital loans to its European allies.
Without the loans in question, British and French economies would've collapsed. London and Paris were thus forced to stop the operation early, and then withdraw from Egypt too.
Recommended reading:
FULLICK, R. & POWELL, G., Suez, the Double War, Hamish Hamilton Ltd., 1979 (there's an expanded and updated 3rd edition by Pen & Sword, issued in 2006)
BURNS, W., Economic Aid and American Policy towards Egypt, (New York, State University of New York Press, 1985)
GREEN, S., Taking Sides: America’s Secret Relations with a Militant Israel, 1948-1967 (London: Faber and Faber Ltd, 1984)
THORNHILL, 'Britain, the United States and the Rise of an Egyptian Leader', English Historical Review, Volume CXIV, Issue 483, September 2004, pp893-894
A very good reconstruction of related US-Egyptian negotiations and relations can also be found in recently-published Wings Over Sinai, by Nicolle, Cooper & AVM Ali Gabr (Helion & Co Ltd., 2017). One of things sticking out from that book is that just a few years earlier, the CIA considered Nasser for its 'asset' in Egypt.