Among the cases she is expected to sit in on when the new term starts in October are two major First Amendment clashes: one involving California’s attempts to limit the sale of violent video games to minors, the other on the free speech rights of protesters at military funerals.

Because of her role as solicitor general in the Obama administration, Ms. Kagan has already identified 11 cases on the docket for the next term in which she would disqualify herself because she had worked on them for the White House. One concerns the privacy rights of scientists and engineers at the Jet Propulsion Laboratory who object to federal background checks.

In the final vote, 5 Republicans joined 56 Democrats and 2 independents in supporting the nomination; 36 Republicans and one Democrat, Senator Ben Nelson of Nebraska, opposed her. In a sign of the import of the moment, senators formally recorded their votes from their desks.

The partisan divide over the nomination illustrated the increasing political polarization of fights over Supreme Court nominees, who in years past were backed by both parties in the absence of some disqualifying factor. Ms. Kagan received fewer Republican votes than Justice Sotomayor, who was supported by nine Republicans in her 68-to-31 confirmation on Aug. 6, 2009. Democrats balked at Samuel A. Alito Jr., nominated by President George W. Bush, with only four endorsing him in a 58-to-42 vote in January 2006.

Most Senate Republicans challenged Ms. Kagan’s nomination until the end, asserting that she lacked sufficient experience and had unfairly stigmatized the military by supporting a bar on recruiters at Harvard Law over the military’s policy against allowing gay men and lesbians to serve openly. They said her record in both Democratic administrations and her strong ties to Mr. Obama suggested that she would try to imprint her own political values and those of the president on court decisions.

“Whether it’s small-claims court or the Supreme Court, Americans expect politics to end at the courtroom door,” said Senator Mitch McConnell of Kentucky, the Republican leader. “Nothing in Elena Kagan’s record suggests that her politics will stop there.”