No, the United States of America is not part of the United Kingdom. They are two entirely separate sovereign nations with distinct governments, legal systems, and histories.
The UK is a sovereign state comprising four constituent countries:
It's a constitutional monarchy with a parliamentary democracy, headed by the monarch (currently King Charles III) and governed from Westminster in London.
The USA is a federal republic of 50 states, a federal district (Washington, D.C.), and several territories. It gained independence from Great Britain in 1776 (recognised 1783) after the American Revolutionary War. It has its own constitution, president, congress, and legal system.
The confusion sometimes arises because:
But politically and legally, they have been completely separate nations for nearly 250 years.