America is Europe, but more precicely, as it was called early on, New Europe.
The way that Americans feel European is that in which we were a re-planting of a European branch in an all-together different territory. There is, still, a pathos of letting be ramifications of Europe that are impossible in the crowded territories of the mother continent, because of space and historical age.
If Americans severed ties with their European forfathers, it is because Europeans from the continent never felt this possibility, America was always simply a mining operation.
Consider Simon Bolivar's independence project. His ambition was to conquer all of America and re-name her Columbia. By taking the naming out of the continent's hands, there was a severing, but by replacing the name with what was considered a more accurate European discoverer of the new Continent a very clear intention and pathos was evident.
Americans can only self-value as an extention of Europeans because, following the mass wipe-out of the people they found there, Europe is all there is. Possibly the one instance in history over the last couple thousand years, at least, where a whole swath of the Earth was really honestly colonized.
The "indians" and blacks here are outsiders, and the mixed (vast majority by now) are as European as the Moors in Spain became (in most cases more, because only a small portion of the American continent ever experienced true culture before Europe).