On Christmas Eve in 2009, Secretary of State Hillary Clinton was awake before dawn to personally monitor a critical moment in the nation’s history.

But Mrs. Clinton, the country’s top diplomat, was not observing a covert operation in the Middle East or tracking pivotal negotiations with a foreign power. Her television was tuned to C-Span, and she was watching the Senate vote on President Obama’s landmark health care law.

Emails released last week by the State Department that were found on Mrs. Clinton’s private server show that she was keenly interested in the administration’s push to win passage of the health care law.