The Declaration of Independence and subsequent creation of the United States destabilized and weakened many Indian nations near the thirteen colonies. The British Empire had largely prevented colonists from moving westward over the Appalachian mountains and into Indian territory. This didn't stop many from trying, and border conflicts between land-hungry colonists and various Indian tribes were common occurrences in almost all of the colonies. The reason the British tried to prevent colonists from expanding into Indian territory was because the British Empire relied on Indian allies for maintaining valuable trade routes through the interior of North America.
When the colonists declared their independence, Jefferson wrote in the Declaration that the British had “endeavoured to bring on the inhabitants of our frontiers, the merciless Indian Savages whose known rule of warfare, is an undistinguished destruction of all ages, sexes and conditions.” Colonists saw British alliances and trade with Indians, who many colonists viewed as savages and enemies, as another form of oppression from a distant king.
Most Indians viewed the American Revolution as, basically, a British Civil War, and remained mostly neutral, with a few tribes siding with the colonists and a few more with the British. But once the Americans secured their independence, they removed all colonial prohibitions on westward expansion. Thousands of colonists began to pour into what is now considered the Midwest, and the native groups there quickly fell to disease and violence. The British no longer had the sovereign power to mediate these conflicts or protect Indians from colonists, so violence ensued. Over the next half century, nearly all Indians east of the Mississippi lost their political independence, saw massive declines in populations (which brought social instability), and lost the lucrative trade with England that was now monopolized by Americans.
Overall, American independence hastened the decline of Indian political independence, destabilized their societies by displacing them from traditional homelands, and ruined valuable commerce with the British that had sustained local Indian economies for a few generations.