What event sparked both the formation of a distinct American identity and a severing of cultural ties to Europeans?
The textbook answers to this are "westward expansion" and "The Revolutionary war"...
...But both of these were gradual processes, not so much isolated events. Frankly, there is more than one correct answer to this question, and I would dispute that America ever "severed" its cultural ties to Europe.
The big difference between Europeans in America and those who never left Europe began with encountering Native Americans (or Indians). For thousands of years, Europeans could conceive of no greater freedom than to choose which of two hereditary tyrants to follow as their king. This was only rarely a choice, and there was never a third option under any circumstance.
Then they met the Indians, who could drop in and out of their society at will and choose to bow to no sovereign. The earliest settlers and colonists' minds were blown at this possibility. Americans had a very different paradigm for what "freedom" meant from their European counterparts.
Another "event" was the literary ascension of Edgar Allen Poe. America had other, earlier literary figures (like Washington Irving and Nathaniel Hawthorne) who wrote with a distinctly American voice, but Poe was the first American writer to develop popular and critical acclaim in Europe, creating a precedent for Twain, Hemingway and Fitzgerald to do the same.