The actual vote to sever ties with Great Britain took place two days prior, on July 2, 1776. John Adams famously believed that July 2nd would be the date celebrated by future generations of Americans, but because the Declaration's printed broadsides were dated July 4, that became the date history remembered.
The famous, beautifully calligraphed parchment document displayed at the National Archives was not created or signed on July 4th. The vast majority of the delegates did not sign the document until nearly a month later, on August 2, 1776.