This is a difficult question because it depends on conceptions of ownership that may differ between cultures. America did steal some Native American land, though it is murky in other areas.
Prior to English colonization of North America, the Native Americans, or at least a portion of them, had come into contact with European explorers. The Europeans carried diseases that the Native Americans had no resistance to, leading to a significant number of infections and deaths. Some have speculated that between 90% and 95% of Native Americans died due to the spread of these diseases before the colonists arrived from England. (A side note here is that the high mortality allowed the unimpeded growth of more trees and other plants in North America, which may have indirectly led to the Little Ice Age in Europe between 1645–1715.) This high mortality rate left large areas of previously settled land, including some villages, completely depopulated. When the colonists arrived, they took advantage of abandoned villages to survive during their first few seasons and later took advantage of previously cultivated lands and pathways to aid in the growth of settlements.
In some areas, such as the Rhode Island colony, the colonists purchased land from local tribes. Whether these purchases were completely valid is difficult to determine, because it depends on the intent of both parties. If Native Americans believed they were accepting payment to allow settlers to use the land, without ceding the right to use the land themselves, then the sales were likely invalid, since the settlers desired exclusive use of the purchased land. It is also unclear whether the group of Native Americans that sold the land could validly represent all the groups that used the land throughout the year. The difficulty of determining whether purchased land was stolen from Native Americans will be compounded with the Louisiana Purchase, as it is unlikely that France had a valid claim to all of the land it sold.
In other areas, especially in the Mid-Atlantic colonies, there were significant hostilities with Native American tribes, at least partially due to white encroachment upon lands that belonged to Native Americans. This problem was exacerbated with colonists pushing over the Appalachian Mountains. In some areas, settlers established settlements among allied tribes, but the encroachment ultimately led to the French and Indian War, which allowed for Native American Tribes to be pushed off of lands they owned if they were killed.
After the founding of America, the theft of Native American land became more obvious. Seminoles were killed and pushed off of land in the acquisition of Florida. Native tribes on the East Coast were forced to move into the middle of the continent, most notably on the Trail of Tears. As white Americans pushed west, lands were taken from Native Americans, either through force or through treaties. The treaties themselves likely count as theft, for two reasons. Firstly, most treaties were forced onto tribes with the threat of violence. Secondly, of the more than 500 treaties the United States entered with Native American tribes between 1778 and 1871, almost all have been violated in some way by the United States.
One reason for this treatment of Native American land was articulated in Johnson v. M'Instosh, which was decided in 1823 before the Supreme Court. In this decision, the Court held that individuals could not purchase land from Native Americans, because Native Americans held only a right of occupancy over their lands, not a right of ownership. The Court stated that ownership of unclaimed land resided with the (white) "discoverer" of the land. Since the Court stripped Native Americans of ownership over the land, it determined that the United States was the discoverer of the land and, therefore, was the only entity that could validly dispose of the land.
In summary, while white America did not explicitly steal all of the Native American lands that it claimed, it did not act in good faith in obtaining any of those lands. Theft by force and theft by deceit are both forms of theft. Morally, America stole the lands of the Native Americans.