The Royal Proclamation led to much resistance, and was one of the causes for seeking independence by the colonies. Even during the revolution, the colonies sought to formulate a coherent policy toward indigenous nations. After achiveing independence, this policy, together with policies for expansion, became codified in laws and the constitution.
Agreements and treaties are signed, and the United States enacts its first laws regulating how to interact with Native nations. In successive acts, it ironically draws heavily from British policies, consolidating the power over such regulations with the federal government, seeking peace instead of conflict, and trying to regulate an orderly expansion through treaty-making. The practices of expansion, of course, did not always follow the intentions of the official documents produced: settlers, land speculators, and others wanted to turn territorial claims into territorial possession, regardless of the means in which it was achieved.
The Louisiana Purchase finally opened the door for future expansion, and, for the time being, eliminated competing European claims for that project.