The idea that Japan's fate was sealed by the attack on Pearl Harbor doesn't relate directly to the atomic bombs. It's difficult to say what exactly is meant without seeing what sources you're reading or studying, but historians (especially more modern studies of World War II) have mostly argued that the Japanese never had a realistic chance of defeating the United States in an all-out war. By drawing the United States into the war via the attack on Pearl Harbor, Japan ensured that the United States would be committed to carry out a long war and would eventually prevail.

Americans armed forces in the Pacific War certainly rallied around the idea of avenging Pearl Harbor throughout the war, but that was never fulfilled in a single moment or action. The Doolittle Raid in 1942 was considered in some ways to be revenge for Pearl Harbor, as a carrier-based air attack on Tokyo. The Battle of Midway, where four of the six aircraft carriers used in the Pearl Harbor attack were sunk, was also often held up as payback for Pearl Harbor. Air attacks on the Japanese anchorage at Truk lagoon were compared to Pearl Harbor. Late-war carrier raids on the Japanese home islands were also billed as similar to the attack on Pearl Harbor.

Mostly, though, the idea that Japan's "fate was sealed" by the attack on Pearl Harbor refer to the warmaking capacity of the United States compared to Japan. Many modern historians believe that there was simply no way the Japanese could have achieved their objectives against the United States in the Pacific and reached a peace agreement. The Battle of Midway is usually considered the "turning point" in the Pacific, after which Japan's naval forces could no longer continue their offensive operations. However, even if Midway had been a complete disaster for the United States -- if all three American carriers were sunk and all four Japanese carriers survived -- it would not have changed the fundamental strategic situation. The United States could replace its losses; Japan could not.

Anthony Tully has an informative breakdown on his website^1. The United States produced 17 Essex-class aircraft carriers during the war, each of which was superior to any of the Japanese carriers built before or during the war (Shinano may have been a fair match, but let's not quibble over such a minor point here). There were also more than 100 light carriers and escort carriers churned out of shipyards. US shipyards launched more destroyers in 1942 alone than Japan constructed during the entire war. This doesn't even get into the construction of merchant shipping, where it can be argued Japan truly lost the war.

Ian W. Toll has a shocking anecdote in "The Conquering Tide" ^2 about Japanese production of the famous Zero fighter plane. There were two main factories turning out Zeroes, one of which was at Nagoya:

When a government inspector passed through the Nagoya works in late 1943, he was surprised to learn that newly manufactured Zeroes were still being hauled away from the plant by teams of oxen. There was no airfield adjoining the Mitsubishi plant. The new units had to be transported overland to Kagamigahara, 24 miles away, where the navy would accept delivery. The aircraft were too delicate to transport on trucks, and the railheads were not convenient. Twenty oxen had died, and the remaining 30 were verging on complete exhaustion. Feed had been obtained on the black market, but the supply was not reliable. Essential wartime deliveries of replacement aircraft thus hung on the fate of a diminishing herd of underfed beasts. Mitsubishi engineers at length discovered that Percheron horses could haul the aircraft to Kagamigahara faster and required less to eat. These ludicrous exertions, when compared at a glance to the arrangements at Boeing, Douglas, or Gumman, tell most the story of Japan's defeat.

Over and over again, you can find clear examples where Japan was outmatched by American resources and industrial might. Japan has always been a net importer of oil; in the late 1930s about 80 percent of Japanese oil imports came from the United States ^3. Other key materials also came largely from the United States, including scrap iron, rubber, and bauxite. The United States imposed wartime rationing on its civilian populations but never seriously wanted for any raw materials. It had all that it needed to produce everything it needed to wage war in Europe and the Pacific, plus extras to produce more materials for the damaged industries of Britain and the Soviet Union. Moreover, American industrial capacity increased significantly throughout the war. This is an older paper^4 but it demonstrates clearly that Japan's economy was already running at near-maximum efficiency by 1941 after four years of war in China. In 1941, Japan produced 5.8 million tons of steel; the number rose modestly to 7.8 million tons in 1943 but fell to 5.9 million tons in 1944; it was about to drop precipitously in 1945. Japan imported less coal every year from 1941-45.

The United States had all the iron, rubber, coal, bauxite, oil, and other materials it needed to produce everything that it needed to wage war, then extra to send to Great Britain and the Soviet Union. Japan had hoped that its early offensive in the Pacific would secure its hold on the resources it needed and had been mostly importing before the war broke out. But this was always a dicey proposition, relying on long, exposed supply lines, and a limited industrial capacity to utilize those resources if they reached Japan.

These realities were recognized and weighed by political and military leadership in the 1940s, but often glossed over in military histories, which may focus on a single battle or campaign. The idea of Japan's "fate being sealed" requires looking at the bigger picture. Winston Churchill's reaction to the attack on Pearl Harbor, recorded in his book "The Grand Alliance"^5 in 1950, is informative:

No American will think it wrong of me if I proclaim that to have the United States at our side was to me the greatest joy. I could not foretell the course of events. I do not pretend to have measured accurately the martial might of Japan, but now at this very moment I knew the United States was in the war, up to the neck and in to the death. So we had won after all! ... How long the war would last or in what fashion it would end, no man could tell, nor did I at this moment care. Once again in our long island history we should emerge, however mauled or mutilated, safe and victorious. We should not be wiped out. Our history would not come to an end. We might not even have to die as individuals. Hitler's fate was sealed, Mussolini's fate was sealed. As for the Japanese, they would be ground to powder. All the rest was merely the proper application of overwhelming force.

The strategy of the attack on Pearl Harbor was to cripple the US Pacific fleet at the outset, preventing a counterattack that could interfere with Japan's attacks on the Philippines, Southeast Asia, the Dutch East Indies, and elsewhere. Japan hoped that destroying the US Pacific fleet would allow it enough time to capture, consolidate, and reinforce these territories to the point where the United States would be unwilling to pay the price to retake them. General outlines expected this process to take about two years, meaning Japan sought to prevent the US from undertaking any kind of counterattack for about two years. Japan hoped the United States would sue for peace in that time, allowing the Japanese empire to keep its newly won territories in the Pacific that would make it economically self-sufficient.

That strategy failed for a number of reasons. For one, the US fleet in the Pacific had sufficient striking power in its aircraft carriers to counter Japanese moves in 1942. The Japanese never had adequate merchant shipping and protection for its supply lines to see this strategy through. Japan's industrial capacity never had the ability to match that of the United States, which was the decisive factor in any protracted conflict. In addition, the American people were so shocked and angered by the attack on Pearl Harbor that the public was galvanized to see the war through. This is what most historians mean when they refer to the idea that Japan "sealed its fate" with the attack on Pearl Harbor.

Sources

1 - "Why Japan Really Lost The War," by Anthony Tully, link

2 - "The Conquering Tide: War in the Pacific Islands 1942-44" by Ian Toll

3 - "Japan’s Energy Conundrum" by Phyllis Genther Yoshida, link

4 - "The Japanese War Economy: 1940-1945" by Jerome Cohen, link

5 - "The Grand Alliance" by Winston Churchill

In addition to those cited inline above, these books also have significant material which helped inform my answer:

"At Dawn We Slept," by Gordon Prange, (on the planning and aftermath of the Pearl Harbor attack)

"Pacific Crucible," by Ian Toll (a prequel to "Conquering Tide")

"The Rising Sun," by John Toland (spends a great deal of time examining Japanese considerations for starting the war)

"Hirohito's War," by Francis Pike (which devotes a bit more attention to pre-war economics than most books about the Pacific War)