How did American settlers justify why they were supposed to control and settle all land to the west?
American settlers justified settlement of the Western United States mostly through the concept of Manifest Destiny. This concept dictated that it was the will of God that the people of the United States spread to the Western edge of the American continent as a prophecy of sorts, so that any question of the motives of such efforts was akin to questioning one's faith. This also played well into the theory of American exceptionalism, by which the special virtues of the American people and their institutions entitled them to expand without consideration of the Native Americans and others already on the land.
In fact it was a special kind of consideration that also justified the expansion, the Old World (the continent as it was) was corrupt and wayward, so by spreading European values and high examples, settlers were in fact saving these groups from themselves...
Despite this, Manifest Destiny is a disputed concept among historians. Even among advocates, there were no agreed upon rules, methods or specific plans to carry out, with some even believing the use of force betrayed the very concept. The Mexican-American War is likely the most famous example of the concept put into action by President James K. Polk as part of his campaign appeal to voters.
Other Democrats largely believed in it but many leaders (Abraham Lincoln) and some political parties (The Whigs) agreed that it was little more than American imperialism, and contrary to the values the United States stood for and were founded upon. Now that could be said to be a high example indeed!