The Senate on Thursday confirmed Judge Sonia Sotomayor as the nation’s first Hispanic Supreme Court justice, concluding a 10-week battle with a resounding victory for the White House.

The largely party-line vote, 68 to 31, brought Judge Sotomayor, 55, to the threshold of one of the United States’ most prestigious institutions, completing an extraordinary narrative arc that began in a Bronx housing project where the Puerto Rican girl ...