I don't think it's likely. New Guinea was a very hard piece of land for colonizers to exploit. Its primary value was as a coaling station, and also for the simple prestige of acquiring land.
This made it valuable to the German colonial lobby who wanted Pacific real estate, and very valuable to Queensland.
To the US, in 1884? They have no shipping lanes to protect, since they don't have their Pacific possessions yet. Nor was there any public appetite for the colony, let alone one hungry enough that the US government would go up against London (and to a lesser extent Berlin and the Australian colonies) for no tangible benefits.
Germany or Queensland might get more or less of the island. The Franco-Italian De Rays expedition might, just might have set up a colony though I doubt it.
But I don't see why the Americans would even want to be in the race.