Patil was elected by federal and state legislators last week and succeeds Abdul Kalam, the popular former president.
Although the office is largely ceremonial, the president has a role in the forming of governments at the state and federal levels. The prime minister wields executive power in India.
The Congress party-led government has billed Patil's appointment as a historic day for India's women, but critics said it was a hollow gesture after a campaign marred by mud-slinging and partisan politics.