Simplest answer is: they couldn't.
The US did not join the First World War when it began in 1914 and most of the German colonial possessions in the Pacific were carved up and captured during the first year of conflict. By the time the US joined the war in 1917 the question of the Pacific had long since been decided. Australian troops had completed the annexation of German territories in Papua New Guinea and neighboring islands, New Zealand had taken control of Samoa, and Japan had seized the German-claimed islands of the Marianas, Carolines, Marshalls, etc (most of the region of Micronesia). At the end of the war, the League of Nations recognized these claims as Mandates-- so they were not territory that the colonial powers could annex, they were territories that they should "tutor" until such time as they could become independent-- though what they means exactly was left pretty nebulous and effectively the islands will be treated as colonial territories, and in the case of Japan Nan-yo, or the South Seas, settler territories.
It's hard to do a what-if, but considering the US had been keen on annexing all of Samoa, not just the Eastern half of the archipelago which became the US territory of American Samoa, it would have made sense for the US to occupy the German half and unite the islands again. Especially as Samoa had only been divided by the US and Germany in 1899. Additionally, the US naval base on Guam was otherwise totally surrounded by German islands that then became Japanese, so expansion makes sense on paper in the abstract. However, like I said at the outset, the US was a neutral party to World War One, so they could not simply attempt to claim Germany territories and remain neutral at the same time and there was never a debate about entering the war simply to claim Pacific islands-- especially when allied governments moved on them swiftly.
As to the Second World War-- there were no major Polynesian Islands under Japanese control over which the US could assert a claim. French Polynesia was/is part of France, Fiji under the UK flag, Samoa under NZ, Tonga in friendship with UK, New Zealand an independent settler state, Hawaii already under a US flag. Further, the United Nations did not permit colonial expansion after the War-- at least not in terms of territorial integration.
After the Second World War the US governed the Trust Territory of the Pacific, which was made up of Japan's former claims in Micronesia. Just like the Mandate system of the League of Nations, the point of governance was preparation for independence. The US did hold onto the islands for a quite a while as they were seen as key strategic locations in the Pacific and an important buffer to counter against the spread of communism-- though in reality the USSR remained relatively aloof of the Pacific and the US found itself facing off against Non-Aligned nations/nationalist movements whose goal was the reject both the US and USSR as potentially destructive belligerents.
Independence came late for many of the islands of Micronesia compared to other territories in the Pacific. The islands had become (and remain in the case of Kwajalein) key US ballistic testing sites, not to mention the atomic testing programs on Bikini and Enewetak. The existence of these programs raised issues as to whether or not the US was preparing the islands for independence or creating a situation wherein they would remain perpetually dependent. A key provision of the Trust Territories was caring for the health and well-being of the people of the territories and that nuclear testing had ever occurred seemed antithetical to that. The US wanted to negotiate favorable political treaties with one super-Micronesian state to basically maintain the status quo. When negotiations over independence began, they were quite heated as the US wanted to be able to deploy military assists within the nations and set up bases. The approach of various negotiators varied, the Marshall Islands were more willing to allow a continued US presence, whereas Palau was not. US negotiators moved to craft separate treaties with different island groups, effectively breaking up the Trust Territory into different nations despite attempts by local leaders to maintain cohesion. The Marianas Islands became the Commonwealth of the Northern Mariana Islands in 1978 (they are a commonwealth territory of the US), The Republic of the Marshall Islands in 1979 which remains closely tied to the US and home to the US installation at Kwajalein, The Federated States of Micronesia in 1979, and then the Republic of Palau in 1981. This was not the end of the politicking between the US and regional leaders, but it was largely the end of territorial administration by the US. By that point there was a lot of international and domestic (Pacific-based, not continental US) pressure to wrap up the Trust Territories.
A few suggested readings on this topic-- mostly related to Micronesia since the history of that region more closely answers the question.
Mark R. Peattie, Nan'yo
Francis X. Hezel, S. J., Strangers in Their Own Land
David Hanlon, Remaking Micronesia